What is Elo?
Your Elo is essentially your rating. It is determined by your win/loss ratio and how many games you have played. In normal queue your Elo is hidden from yourself and others, but the matchmaking system still uses it to decide your opponents. In ranked games your Elo is there for the world to see.
Contrary to popular belief, "ELO" isn't an acronym nor was it invented by Riot. The Elo rating system was made by Arpad Elo, who named the system after himself. It was originally used as (and still is used as) a rating system for chess players, and will most likely use exactly the same algorithm to calculate their skill level, though it will be a close variant if not exactly the same. You can read more about Elo and its creator here upon its Wikipedia article.