(Note: Newbie is not meant as an insulting term. Merely as a way of identifying the players who are new and/or need help with the game)
1. Teamplay > Individual Efforts
If you learn one thing, learn that all members on your team have to work as a team and communicate their motives. Work as a unit and things will come much easier. The most skilled players can lose if their team doesn't coordinate, and the least skilled players can win if their team coordinates. Ping the minimap when you want to gank (kill) someone, when you see one of the enemy champions alone and ready to kill, when you want to get somebody's attention about an empty lane
2. Know Your Champion's Role
There are different types of champions in this game: Support, Assassin/Ganker, Carry, and Pusher. How they achieve these things differs from one champion to the next, but these are the most fundamental archetypes that all the champions fit into (in varying degrees, of course), with some being better at one thing than another or some fitting into multiple roles. Learn these roles and play to them. Generally you'll want to have at least one of each in your team, but it isn't impossible to win otherwise.
Ganker/Assassin - You'll spend most of your time hunting and killing enemy champions.
Carry - You'll generally be weaker in the early game, but late game your champion shines with the right item choices.
Pusher - You have skills that make it easier for you to either take down towers or kill many minions at once.
Support - You support other heroes by healing them, giving them passive auras, or helping them achieve their class roles in one way or another.
3. ALWAYS Watch The Minimap
The worst thing you can do for your team is die. It gives the opposing team money and experience that can quickly snowball into a serious advantage for them. Thus, by default the worst thing that you can do is to let enemy champions out of your sight because they will pop up somewhere and quickly overrun your single champion and killing them. When you notice that the enemy champion in the lane you were in is missing, call out "missing <champion name>" or something similar so that other people are also aware of the situation. This goes back to working as a team and communicating relevant information.
4. Don't Stop Leveling
Another mistake new players make is not taking advantage of empty lanes full of juicy minions (EXP and GOLD!). I've seen many games in League of Legends won simply because of one or two champions that were allowed to farm minions in a lane without any competition or threat. This is a big mistake that can be remedied by not allowing any lanes to remain empty for very long (sometimes it's necessary so that you can gank, kill neutral minions like the dragon or Baron, or go back to base to heal and buy items). Don't have your team all go into the same lane because experience from minions is shared. A tip that ties into this one is to always go for last hits on minions so that you earn as much gold as possible (another reason the neutral creeps are good!).
Don't stop leveling and earning gold!
5. Focus On Your Champion's Strengths
If you're playing as Ryze, you know that his strength is in his nukes and his root and he has the archetype of ganker. You'll know that something like lifesteal, armor, or hit points aren't exactly his forté. You should stack his ability power to make his nukes stronger so that you can gank other champs much easier. If you build him with armor and attack speed, for example, you'll just be an average nuker with a fast attack speed. Not very scary. That isn't to say that you should ignore other item stats completely, but that you should definitely focus on one or two things that play to your champion's strengths. Learn your champion's strengths, focus them, and own some face!
6. Know Your Items
As Drastone said below, it is important to know which items are good for the situation that you find yourself in. Going against a team composed of close-range physical damage? Try a Sunfire Cape. Lots of mages? How about Wit's End, or Banshee's Veil? Also something that doesn't get across to many people is that an item property with "UNIQUE: <item> does X" means that you don't get any additional benefit from stacking that item. Generally speaking, don't ever take two of the same items with UNIQUE.
Know your items. This comes with practice, more than anything, but the League of Legends website was recently updated with a slick item list that you can browse at any time. If you notice someone absolutely dominating the game, sometimes a good idea is to look at the items they're using and learn from them, also.
If anyone has any other suggestions for newbie tips, please do post them. Keep things concise and relevant, if you can, so that newbies can look at these tips and absorb them quickly. Nobody wants to read a f!@#ing encyclopedia.
