Q: How does Elo work? What does it mean? Where can I see it?
Elo is the ranking system used in LoL. The way is works is that when you get matched up against a team, your team will be expected to win or expected to lose, depending on your Elo.
Ex.
Team A - 1250 Elo Ranking
Team B - 1350 Elo Ranking
In this scenario, Team B is expected to win. If they win, they will gain less Elo because they were the stronger team going in.
Team A - 1250 -> 1242
Team B - 1350 -> 1358
However, should Team A win, they will gain more points than usual because it was an upset.
Team A - 1250 -> 1263
Team B - 1350 -> 1337
Note that this is acting as if the entire team has their own Elo, what actually happens is that a team's Elo is considered depending upon the players inside and other factors. As an example:
A pre-made comes in with these players:
Noob - 1000
N0ob - 1000
Noobie - 1000
Newbie - 1000
Newskis - 1000
Now, this team doesn't have a 1000 point rating when it gets put into the pool. What happens is that they get a slight modification (between 25 and 200 points, I think) because they are a premade, and are expected to be stronger than simple pubs. So, even though all the players are 1000 ranked, the actual team would probably be ranked 1100-1150.
There isn't any way to see your ELO ranking from Normal games. The idea there is that the matchmaking is for fun, and that you shouldn't be worried about losing. When Ranked games come in, those are where the ladders and competitive play will be shown.
ELO Wikipedia (Link)
ELO Calculator (Link) *Note* The ELO Calculator isn't 100% accurate, it can't calculate leaves in it, but it should give you an idea of your estimated ELO.