It's a PvP game that revolves around tactic, teamplay, careful planning and good reflexes. Each player controls 1 of over 100 unique champions. 2 teams of 5 players are trying to destroy the other team's base. Each team's base sends out waves of minions over three lanes. This is the map: [link] It might be a little out of date but the essentials are still the same. How to win a game: Destroy the enemy's Nexus. Before you can do that, you have to destroy the 2 heavy towers in front of it Before you can do that, you have to destroy one of the 3 Inhibitors. Once an Inhibitor is destroyed, the base spawns a really heavy minion with every wave that goes on the destroyed Inhibitiors lane. But before you can destroy any Inhibitor, you have to destroy the 3 Towers before them. And you have to destroy the towers in order too. This might clarify where the lanes and their towers are: [link]
Here's the homepage if you're interested.
I don't know if you know Warcraft3, but this game actually started out as a very popular mod for it, named DotA (Defence of the Ancients).
And here is the reason I got reminded of it: [link] This little guy is Fizz, the Tidal Trickster. I think he looks lke he could be related to the guy you drew.