Valuable resources:
http://www.leagueoflegends.com/learn/new_user_guide
and all of the stickied threads here: http://www.leagueoflegends.com/board/forumdisplay.php?f=29
especially
http://www.leagueoflegends.com/board/showthread.php?t=194210
http://www.leagueoflegends.com/board/showthread.php?t=110065

You start in your base. There are three lanes: one straight down the middle of the map and two that curve off to the sides.
Usually, one person will take the middle lane and two people will take the other two lanes. Sometimes, one hero will actually not take a lane but "jungle", or kill the creeps in the jungle. This has the advantage of giving a second person on your team a solo lane, meaning more gold and experience for them.

A wave of minions will spawn at the nexus and begin moving down each of the lanes. They attack anything in their path, but will default to enemy minions The first six levels, usually, are spent trying have to go now. If there's interest, I'll finish this later.

-Persistent out-of-game customizable options - 30 "runes" and 30 masteries you can use to boost certain abilities.
-Abilities are more frequently usable due to lower cooldowns, lower costs, and higher mana regen.
-The strategy is more than just "let your carry farm up, AoE stun their team, watch them die"
-No denying. While some complain that this removes a significant amount of skill from the game, it means that lane control is no longer absolutely necessary. You can still zone your opponents out of experience range or at the very least stop them from getting any gold if you have a strong lane combination, but it's not absolutely vital.
-Similarly, games with random people are less frustrating because you don't have to worry about your team having two melee heroes in one lane.
-Spellcasters have Ability Power, allowing them to scale with items instead of just levels. No more do you have to go 15-0 as Lina but still get wtfpwned by their fed Faceless Void because you do nothing late game!
-The more aggressive lane game means early kills are much more common. Particularly one-sided games often end in 25 minutes - half of a standard DotA game. I've played a game with a push-centric team where we won in 18 minutes. Of course, on the other hand sometimes the game is excruciatingly close and lasts quite a while. The longest I've heard of was an 82 minute thriller that ended because one person got caught out of position trying to farm.
-The game isn't bound by the WarCraft 3 engine, meaning many gameplay improvements can be made. Items no longer have to be bought in parts, hotkeys are much simpler to learn (QWER for skills, 1-6 for items, and DF for summoner spells), and you can buy items while dead, to name a few.
-DOCTOR MUNDOOOO!

There are purchasable elements to the game. IP (influence points) are earned by playing games, and RP (Riot Points) are bought with money. However, RIOT promises that no competetive advantages will be purchasable with RP. Here's a simple visual:
Can be bought with:    IP       RP
__________________________
Runes                         YES     NO
Champions                   YES    YES
Skins                           NO     YES
Runepages                   YES    YES


A quick explanation of each:
Runes give basic stat boosts (more speed, more health, ignore some of the enemy's armor). There's a large difference between a full runepage and no runes at all, but generally if you don't have a full page it won't be a big deal. For instance, a full set of flat health quintessences give you an extra 100 health to start out with, on top of a base 450-700. You can, with your starting money, purchase an item that gives that much health, some health regeneration, and some armor.

There is a selection of 10 free champions that rotates each week, giving everyone a chance to try new heroes out before committing to a purchase. Buying heroes costs (approximately) 5 games for the cheapest up to 60+ for the most expensive. With real money it costs either 20 dollars for a starter pack with 20-some heroes, or $1 - $10 depending on the hero. Most people suggest playing free heroes and building up IP until you find one you really like, then purchasing and playing him. I loved Doctor Mundo, so I bought him and him for some 250 games with short breaks to try free heroes. Right now I'm playing mostly Olaf.

Skins are purely cosmetic. They range in price from about 2 bucks for a simple palette swap to almost 20 for a hilariously awesome concept, new animations and a whole new voice set. RIOT makes most of its money here.

Runepages are a new feature. Previously you were limited to just 2 pages of runes, which was problematic for people that liked to play lots of different heroes. Now, you can have up to 9 I believe - more than enough to cover most character archetypes. Of course, you have to pay for these extra pages, and they aren't cheap. One page is as much as the most expensive champion, and the 7 page bundle costs almost 30 dollars. Still, they are far from necessary to win, and two allows for a "Generic Physical" page and a "Generic Caster" page, or one page for your favorite hero and one all-purpose page.

http://img838.imageshack.us/i/battlez.png/
http://img26.imageshack.us/i/screen31n.png/
http://img835.imageshack.us/i/cleavered.png/
These are all versus bots. Once the server is back online, I'll work on getting some nice screenshots of an actual game.
I'll also work on getting higher quality ones.

Here are three from close to the end of a real game.