ELO is a global ranking system. It was first used in chess competitions.. 1200 elo rating is the average, while above 2000 are masters. (see http://en.wikipedia.org/wiki/Elo_rating_system)
On that game though, Elo is quite weird, since this ranking system was made for individual skills ranking (chess, starcraft, any 1v1 game). Though, this is a team game.
Anyhow, to gain Elo you must play ranked games.. If you win, you gain 15-20 elo and LP (League points are NOT related to Elo at all!), if you loose games, you loose elo acordingly. It is that simple.
League points though are a little misleading, since those ranks ("bronze, silver, gold, plat, diamond") actually are fake. You always have your "hidden elo" which is your real rank.
Little tip : Don't be scared to dodge games (close the LOL client while in team select) since your MMR (hidden elo) doesn't get hindered by that, though your LP while drop by 5 LP or 10 LP. But it doesn't really matter since your LP gain is based on your MMR.
Sorry about the random rambling, but that quite sums it, if you have precise questions, you can ask them to me in a reply and I'll try to answer them to my best knowledge.
Sifu