League of Legends is a free to play PvP arena game in the style of DotA, otherwise known as a 'MOBA' if you're keeping up with the acronyms the kids are using nowadays. Being free to play, the major barrier to entry is the crippling fear of entering a game and having no idea what to do, resulting in some 12 year old kid hurling various expletives in your direction and causing your e-peen to shrivel up into a ball and cry.
Luckily though, I am here to tell you how to not make a complete embarrassment of yourself.
Character selection: When you play solo queue (unranked matches - you will not earn the privilege to play ranked matches until you've got a couple of hundred games under your belt) you will find that most people instantly lock in whichever character they feel like playing with no regard for what may help the team towards victory. So, you can dramatically increase your odds of winning by plugging up holes in the lineup.
The most important thing is to make sure your team has a "tank". I use quotes because this is just a label - the other team is not going to actually be beating on the "tank" unless they are stupid. Rather, the tank has enough survivability to deter being beaten on, allowing him to get up close and use skills that soften up or disable key characters on the other team.
An easy tank to play that is considered very effective even in top ranked PvP is Rammus, the armordillo (has naturally high armor as you can imagine). For a dirt-cheap option you can try Nunu the Yeti. My personal favourite is Cho'Gath, who can eat things to grow ginormous (including other players).
The next most important thing that I see commonly left out of team lineups is a ranged physical character. This character's main job is to let the team push down turrets, the automated defense towers that regularly dot the path towards the enemy's base. Without a ranged physical character, your team will end up milling around just outside tower range, scratching their heads and wondering why they can't push anything down, until the enemy gets their act together and wipes them. Your secondary job playing this character is to farm up an enormous amount of minions, getting you enough gold so that if the game goes long you will become an unstoppable DPS machine.
The best ranged physical character to recommend is Ashe. She's considered 'top tier', is cheap, you get to try her in the tutorial, she has one of the best initiating skills in the game, and she's pretty easy to play.
Next, the team will need to have a magical caster. These have high burst damage and are important for drilling down squishy types on the other team. These seem not to be in short supply in the general population. A popular past-time for the common or garden variety noobs is to lock in as many of these squishy characters as possible within the first two seconds of loading into the lobby screen. But if by chance nobody locks one in, you should go right ahead.
Laning vs jungling: The characters mentioned above will often be 'laning'. The most commonly played map is a 5v5 setting, with 3 lanes. An endless army of minions spawn from each base and stroll down one of these 3 lanes, striving to reach the enemy base, but inevitably being converted into gold/XP by hungrily waiting players on the other team. But the maths doesn't quite add up - with only 3 lanes, we will have one person soloing and two people in each of the other lanes.
The solo player levels up a lot faster because they don't have to share XP. So it's desirable to maximize the number of people soloing. To gain an additional solo spot, one player often levels up instead in the 'jungle'. This is imaginatively referred to as 'jungling'. This player moves around farming a gradually respawning series of neutral mobs that stand around in between the 3 lanes. You will note the positions of these mobs by little skull and crossbones symbols on the minimap. This player will periodically emerge from the jungle to gank unsuspecting players.
I don't recommend jungling in your first few games because it's quite easy to die and thus defeat the purpose of this post, which is to prevent you from embarrassing yourself. So wait until you've seen somebody else do it. When starting out I'd recommend that the easiest/fastest character to jungle with is Warwick, the werewolf, because of his combination of damage, healing, and gank-oriented skills. Whoever you jungle with, bring Smite from the collection of summoner skills to ensure that you remain high enough in level to succeed in ganks. Typical starting items are Cloth armor and 5 health potions.
If your team composition has one of each of the above when in solo queue, it will be a small miracle and you will be well on your way to a cruisy win.
Carrys: The players who are given solo lanes will level up faster and get more gold from minions. These players then become your most valuable assets - help them, protect them, and they will "carry" the team to victory.
Summoner skills: Every player chooses 2 from a pool of common skills regardless of your character. For mana-hungry characters I often take Ghost (run buff) and Clarity (mana recharge). Ignite or Exhaust are good on offensive characters. Flash is a nice escape skill on any character. Teleport is useful if you have a solo lane and can't afford to return to base to heal and then run all the way back (since you'll probably lose your tower if you do).
Communication: When you are fighting against the enemy and they disappear from your view, they might be running off to go and surprise your unsuspecting team mates in another lane. If you type 'mia' (Missing In Action) into the chat when they disappear, you will absolve yourself of your minimal obligations, preventing people from raging at you when they die (actually they will still rage at you, but at least you'll know it's not your fault). You can give additional warning by clicking on the minimap ('pinging'). When the player you thought was missing in action returns, type 're'.
Items: There are recommended items for each champion which will get you by in your first couple of games. If you want to stray from these, take note of items that particularly effective players on either team seem to dominate with. You can view items on members of both teams by pressing TAB (this only updates for enemies when someone on your team has seen their character since their last purchase). You'll pretty soon build up a good idea of which sorts of items are good for which sorts of characters.
Brush: Maps are dotted with brush that you can hide in. When in the brush, players outside the brush can not see you. This can lead to comical ambushes where the entire enemy team comes charging out of the brush like clowns from a tiny car. Some skills are able to be aimed at brush and will make an audible sound if they connect with another player. If you don't have a brush-checking skill such as this and you are not sure where the players in your lane have gotten to, they are probably hiding in the brush.
Important neutral monsters: Several monsters around the map give very useful benefits when you kill them. These monsters always spawn in the same location. 'Blue' or 'Golem' refers to a buff that gives huge mana regeneration and cooldown duration. 'Red' or 'Lizard' adds damage and slowdown to attacks. The Dragon gives a nice bonus in gold to the entire team. The Baron is the most difficult and is generally not attempted until your team is level 15+ - it gives a massive bonus to damage and stats, and is usually taken by the winning team to allow them to steamroll and win the game.
Wards: Wards are cheap and ridiculously effective. New players never buy them because they don't know where to place them. Useful places include the Dragon and Baron mentioned above. Generally if you want to push a tower, placing a ward in the most likely place that a gank will come from is a good idea too. Pink wards can see other wards, and they can see invisible players.
Alright, that oughta do for now. Off with ye, go forth and perhaps now avoid complete and utter failure :D