The game is f2p. Every week there are "free champions" that are rotated. You can play those. Every game you get a currency called IP. With it you can buy champions permanently, as well as runes to help out your character. Also you can pay micro transactions for riot points to buy champions/skins/ip/exp boosts.
You have a professions system of 30 levels for your "summoner" profile. It gives you small boosts to your stats.
Each league of legends game lasts about 30-45 min, with your champion growing from level 1-18 each time, buying new items for that round. This all resets every game. Just like DoTA