Where does the game League of Legends take place?

World of Warcraft = Azeroth

Heroes of Newerth = Newerth

Where does League of Legends play out?

In lore, what is this place called?

The setting of League of Legends is on the world of Runeterra, specifically the continent of Valoran. There are two primary city-states, Demacia and Noxus who were constantly at war. The League of Legends was formed in order to control the conflict.

For more detailed information, you can read through the Journal of Justice, which is a sort of ingame newspaper that chronicles goings-on in Runeterra from an in-character perspective, talking about the various Champions, events, etc.

If you'd like a super-quick crash course, you can watch Total Biscuit's LoL Lore-In-A-Minute.

League of Legends takes place in Valoran, an island-based "continent" (if you will) with several large subdivisions, including the two major rival cities, Demacia and Noxus. Valoran exists in the realm of Runeterra.

The actual battlegrounds take place in the following subdomains:

• Summoner's Rift (5v5)
• The Twisted Treeline (3v3)
• The Crystal Scar (Dominion)

For more specifics, the wiki has a great lore entry on Valoran here