Let me preface this by saying I am not the best player around. I play mostly normals since I like to queue with 3-4 friends, but I am about 60 wins above losses and many of those wins have come playing Nidalee. 

Nidalee can be played two ways, Hybrid and AP. AP is usually the par for course since your cougar is too fragile to use late-game, while your heal has an excellent ratio and a great steroid effect for your physical carry. The nice thing is that Hybrid and AP share the exact same items all the way through mid-game, so you sacrifice no ganking power early by going AP late. This guide is the most common build I use but there are definitely variations you can try. Keep in mind that I do not recommend picking Nidalee unless you can guarantee a Solo mid or 1 vs 2 lane. As without the extra levels Nidalee is rendered much less effective mid-game where you need to be causing mayhem, and another champion might be a better choice.

Masteries:
9/0/21 - No exceptions. The entire utility tree provides incredible benefits to Nidalee, and with CDR blues you can hit 15% CDR reduction which will cap you out with blue buff. The only decision you have to make is whether you want 3% crit or 0.60 AP per level, which is just personal preference. I also recommend Good Hands over perseverance as 5-10HP a minute is not as useful as a lowered death timer.

Runes:
Armor Pen Reds
Mana Regen/Level Yellows
Flat CDR Blues
HP Quints

Pretty standard. You take armor pen over MPEN because your spear will hit like a truck if you land it regardless of 9.5 more or less MPEN (a big if on landing it against a skilled opponent though). Whereas without some armor pen to add bite, your early game harass will feel like the gentle caress of a lover to that Xin Zhao using your lane to free farm. 

Summoner Spells:

You must have one of either Flash or Ghost. As long as you have some way to gain some distance quickly your passive and pounce will do the rest. You can take both and be as elusive as bigfoot but that's rather overkill. I usually take flash because you can usually pounce toward a wall and flash into the jungle never to be seen again. Plus flash+takedown is a potent finisher on runners. But it's really personal preference here.

As your second skill you usually choose from:
Exhaust: A good choice since you lack natural CC. Using this on someone after you've hit 6 often results in a kill.
Teleport: Always a useful skill, especially if your team is warding. Also capable of incredible cheese late game with guardian angel. As TP'ing before you 'die' with GA has the same effect as TP'ing with Anivias Egg.
Clairvoyance: A surprisingly decent choice on Nidalee. It can help you steal buffs with Javelin Toss and net kills on runners with the toss as well. Never underestimate the eye of Mordor, especially because if you don't carry it most likely noone will.

I personally take exhaust if our team has 1 or no exhausts, to put on their olaf or kog'maw or what have you. But usually there is no shortage of exhausts so I go with teleport. If you are running in a premade clairvoyance might be your best bet unless you have a Janna or Taric willing to get it instead.

Skills:
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Prowl

- Nidalee's passive seems lackluster, but combined with pounce and your cougar movement speed it makes you one of the best jukers in the game and especially hard to catch unless they blow a lot of hard CC on you. When running away never go down the lane, always go into the jungle where you can pounce into brush and gain movement speed. Even if the shortest path to safety is straight down the lane, you are often better served swerving into the jungle unless you are extremely close to allies or a tower.

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Javelin Toss/Takedown

Her bread and butter skill early. Javelin toss does an obscene amount of damage if you hit with it, but it is easily dodged and blocked by minions. It also has an extremely finicky hitbox like Morgana's dark binding. You will rarely hit with this if you are mid and the enemy is competent, but you can land a few if you are in a 1 vs 2 side lane. Best bet is to hit with it by launching it from the fog or brush. Launching it across the cliffs overlooking the river into the lane when you are close to your tower often gives you a good chance to land a satisfying crunch that chunks Veigar for half his life like your spear is an embodiment of your hand flying over to slap him in the face. Javelin Toss is also great for hitting people trying to recall behind their tower or stealing neutral buffs. Often it is a good idea to ward the enemy blue. Your level 5 spear hits as hard as smite on the neutral creep and you can launch it from the safety of the other side of the universe and walk back to your lane with blue buff and a raging ******. You might not want to hit them with too many spears if you are trying to get a kill, as landing a particularly devastating spear will leave them with the Boomerang Blade effect. Where they have burning paranoia and will refuse to leave their tower after being brought to 40% HP with a single skill.

Takedown is one of the main reasons why you are so feared mid game. It does a rather massive amount of damage if you have been leveling Javelin Toss and can often lead to surprise buttseks kills in the mid lane out of nowhere with your RWEQ combo. Takedown with Sheen and Manamune often hits as hard as Demacian justice as a finisher until the game crawls past the 20-25 minute mark

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Bushwhack/Pounce 

One of the biggest 1 point wonders in the game. I recommend putting a point into it as your first skill then ignoring it until the rest of your skills are maxed. Spam it everywhere since it costs very little mana. Common practices include:
- Run up and down the river before the game starts spamming it everywhere to stop an early gank.
- Spam it in all the brush by you. Bushwhack is also great for throwing into the tip of brushes where that wild garen is waiting while trying to regen his health.
- Spam it in all the buff camps, and especially dragon.

With proper brushwhacking you are very hard to gank. And an enemy tripping one of these will often be forced to do the bushwhack dance where they spin in circles for about 10 years until the sight effect wears off. Your traps are near impossible to see in brush, and are unkillable so often they can't avoid going over them anyways if you place them right. However they don't replace proper warding, they just augment them.

In late game teamfights the delay effect and rather pitiful effect radius makes offensive use of bushwhack near impossible. The best bet I've found is to just put it in front of your squishy carry. As a 40% MR/Armor Debuff is huge on that garen who tries to run up to your Miss Fortune screaming Demacia.

Your pounce is what makes you so hard to catch. It keeps your guinsoo's charged, it carries you from lane to lane fast, and it's a great chasing tool. The only thing to remember with pounce is that you can leap over some walls with it but not others. Like the cliffs overlooking the river by the side lanes, or the little ramps to either side of the mid lane that lead out of the river. Could save your life.

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Primal Surge/Swipe

Not much to say about swipe. It really helps you with your farming, and adds decent damage. But it's pretty hard to miss with it as the radius of effect is surprisingly large.

Primal Surge has a great AP ratio and makes you an excellent healbot late game. Just a guinsoo's is usually enough for this to be healing like a pro, and any more is gravy. The attack speed bonus makes you drop towers incredibly fast if you can cheese your way into an unattended lane, but when the teamfights start happening don't be greedy with it. It's the most effective on a champion with great damage but low attack speed. Ashe is a great choice since she often hits like Mike Tyson once she gets her IE but will be lacking in attack speed. That warwick with madreds first is also not a bad option. Or that Miss fortune who got manamune/brutalizer/BF sword. Just put this where it matters and it will be extremely effective.

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Cougar: 
Not much to say on this. Except to spam it back and forth if you have to if you have on use effects to proc. The movement speed is nice and means you can stick with boots 1 and often still be as fast as your enemies when you take into account your passive.
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My personal skilling order is 1 level in BushWhack. Then 1 point each in spear and heal. Then level spear after that only stopping to pick up your ultimate to make takedown as potent as possible. If you are in an omgwtfbbq tough lane then it might be smart to level heal instead, but really as long as you have the solo you are relatively hard to chase out of the lane with just HP pots and 1 level of heal. Especially since once you hit 6 and get boots you can pounce your way back to your lane from town in no time flat. Just make sure to have at least 1 point in everything by 6 so you have access to all your cougar skills, and only stay with 1 point in bushwhack until end game because the duration is the same no matter what.

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Item Build:
Start with Sapphire Crystal+HP Pots
First trip back make Tear of the Goddess + Level 1 boots + Pots/Wards
Second Trip back Make Manamune and start on your Sheen
From then on finish your sheen, then start on your rageblade.
Work on your mercury treads with any extra gold you have every trip back.
Get a fort pot if survivability is an issue mid game, do not invest in survivability items.

I start with the crystal and HP pots because really as Nidalee you won't be killed unless you monumentally screw up. You have gank warnings from your traps, you have your flash/ghost, you have a heal and pots, and at 6 you have pounce spam. First trip back you should easily have enough for Tear+boots+pots and maybe a ward or two. Warding enemy blue is important if you can. The tear synergizes AMAZINGLY well on Nidalee. Your cougar skills have 0 mana cost, have negligible cooldowns, can be spammed even while you are traveling from place to place, and ALL proc the tears effect. With a single tear and your runes/Masteries you will rarely have mana problems again. It will proc you 80 mana a minute just from spamming cougar skills, which means this 995g item is giving you effectively 23 mana/5 regen in the lane. You can fill this easier than any other champ because you have the golden ticket combination of both no-mana cost and mana cost skills. Meaning it's easy to max without being useless.

After that finish your manamune and work on sheen, although if you've gotten some kills you can often get sheen outright since you are very good at farming. This combination is extremely cheap, can be gotten fast, and will put your burst at amazing levels. It's not uncommon for your takedown to hit harder than most ultimates. Manamune is amazing on Nidalee because it offers damage when she most needs it, solves all her mana problems, and you will be filling the 1000 mana cap at lightning speed. Sheen makes Manamune stronger and adds to your burst damage in cougar form. Which just these 2 and boots you can 1 vs 1 most champions in the game and gank easily. Especially if you pick up red (very easily done by nidalee once you hit level 7/8 or higher).

Afterwards get rageblade because of it's cost efficiency, adding in merc treads when you can. Don't get tabi or swiftness because your pounce lets you flee in terror already. I get rageblade slightly later than usual because the attack speed bonus doesn't help you much in ganks (your ganks are usually bursting them down, not sustained whacks like Olaf). And the AP is more useful mid-late game while the damage from Manamune/Sheen is more useful early when you can get your kills. You can keep rageblade 8 stacked all the time easily unless you are braindead, and this item alone will make your heals and spears very potent.

The benefit of Nidalee is that your core item build of Manamune/Sheen/Treads/Rageblade is both cheap, effective, and versatile. You can finish it fast and get maximum use out of each piece. After this your item choice depends on your situation. Zhonya's Ring is always good, but is expensive to make and can be difficult to finish if you are constantly team fighting. Lichbane is usually your next item since all of it's effects synergize very well on you, and you have your sheen already. You are a good choice to carry Abyssal Specter or Aegis as well if your team needs them. By this time you will be way in the back throwing spears and traps and heals, so the MR from your Lichbane+treads+aegis or specter makes you very hard to kill at a range. And only a foolish Physical Carry would focus you in a teamfight since you can kite them all day and your team will bend theirs over backwards in the meantime. Red elixir is a must mid game if you are having trouble surviving, but switch to blue elixir end game for more potent and numerous primal surges. End game your surge adds way more effective health and DPS than Janna's shield on an identical cooldown, so take advantage of it

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Playing
Level 1-5: Bushwhack everywhere. Sneak in some harassing attacks if you can but play passive. Unless your enemy is herpderp they won't be hit by spear, and your spear is not that devastating low level anyways. If they push your tower hard heal yourself to clear the creep faster with the bonus attack speed. Focus on farming only as if you where a carry.

Level 6-8: You can start being more aggressive now. Cougar lets you farm like a boss and Pounce/Swipe/Takedown will often take off 50% or more of an opponents HP instantly and have very short cooldowns. If you land a spear now they will be in prime life range to be burst combo'd to death by you. If they are a weak pusher (like Vlad) or if you manage to kill them, go take blue or red buff then blue pill home then pounce spam back into lane. You can easily take these as Nidalee with a healing potion or sufficient health without any help and you will be a huge threat both ganking and in the lane. After you reach 6 you are near impossible to kill except by Burst/Ignite combos like Vlad or Annies so you are really reaching your golden age.

Level 9-14: this is when you are dominant. finish your manamune/Sheen and your attacks will be doing 200+ damage with takedown often hitting for 400+. Pop a red pot and go to town. You can easily roam their jungle laying traps and stealing buffs as long as you keep a sharp eye out on the map to see where most of them are. If you have red buff flashing or ghosting in from the river and pouncing on them is almost always a kill unless they blow a summoner spell. If you can get a hard stun to initiate from Taric or Sion or such it's pretty much a guaranteed kill as you can take any non-tank from 100 to 0 in seconds. In teamfights you are still a threat. After chucking your spear/trap/heal immediately cougar and go after their squishy targets. Flash out if you are in trouble or after you've slain them and go back to the usual spear/trap/heal routine

Level 15+: This is when teamfights start deciding games. If you get wiped that means towers and likely an inhib down. Strafe in the back healing your carry and throwing spears with wild abandon. If you have the teamfight won go in with pounce and takedown the runners so you can push.

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Random tips:
- If you flash backwards after shooting a spear it will do additional damage. I've only done this a couple of times but it is kind of funny when it happens.
- If you come out of cougar and immediately spear you will sometimes throw it behind you, an irritating bug that you might have to compensate for. 
- Late game use your spear to check brushes and not your traps, as your spear has infinity and beyond range and makes a huge sound effect if it hits.
- You can push lanes as hard as Sivir since your cougar QWE clears a wave near instantly and your heal lets you really put out the damage. If you find your team in one of those mexican standoffs in the mid you can slink off to the side and quickly push a lane. The combination of your traps, flash/ghost, ranged attack and pounce make this a pretty safe endeavor and sometimes the enemy team won't even notice.
- Don't be afraid to get a snowball item or build your sheen into a trinity force if you are really rickrolling the enemy team. Don't let up the pressure if you have an advantageous situation. 
- You can farm the enemy wraith camp once you are level 7/8 from mid by occasionally chucking a spear in that general direction. You will usually kill 1 every spear shot. and get free money. If you are really savvy you can even slightly leave the lane and farm the small golems in the blue buff camps for money in the same way, but oftentimes this isn't worth it except for the lulz.
-More to come hopefully