Your goal is to destroy the enemy base (the Nexus) by first eliminating the turrets that defend it, which usually requires a heap-load of Cannon Fodder Mooks to tank the turret blasts while the Champions whittle it down.
There are also a number of secondary objectives scattered around the map which grant significant bonuses to whoever claims them.
The numerous memes spawned by the community can be found here.
The numerous Shout Outs to other medias can be found here.
From April 2016 onward, the most popular modes are featured in a special Rotating Game Mode queue, where it is available for play during the weekend.