The way the system works is like this:
There are 7 "Leagues". They are, in ascending order, Bronze, Silver, Gold, Platinum, Diamond, Master, Challenger.
Each League, excluding Master and Challenger, is separated into 5 Divisions. In order to go from one League to the next, you must rank up through the Divisions.
The way that happens is through playing Ranked. Your first ten games of Ranked, every season, are your placements. After those ten games, you get placed in a certain Division, in a certain League.
The way you rank up through Divisions, is reaching 100 points, then qualifying for your promos. You get a certain number of points for winning, and lose a certain amount for losing.
What people mean by the term Elo, is the place your currently at. For example, Diamond 4 Elo, or Silver 3 Elo. They are really just rough terms to describe how far up the ladder you are.
Now, to actually answer your questions:
You can find your Rank on various sites, Lolking for example, or you can open the client, and go your Summoner Profile. There you'll find your Summoner Name, Champion Mastery, Match History, etc. In the bottom, right-hand, corner, you'll find a description of your current rank, and a number from 0-100. If you haven't done your promos yet, there will be a grayed out area, stating x/10 games complete.
Every time you end a Ranked game, the end screen will show your grade/champ mastery increase. Below that is how much you gained/lost from that game.
Ranked is the only way to gain Elo, but your Elo is different for each Ranked Que, Flex, TT, and Solo/Duo.
