In a Rogue Run and in Endless, every fifth wave is a boss wave.
Three bosses
The first boss is always the classic: a shielded block that sweeps across the board. Later bosses can be the Fissured one, which splits in two, or the Warden, which teleports and walls itself in. Each has its own health bar.
Fifteen relics
After a boss in a Rogue Run, you choose a relic: a lasting effect for the rest of the run, such as Overcharge Core, Swarm Core or Midas Core. You pick one of two; some relics unlock as you play.
Then the fusion
A boss wave is also when you can fuse two mastered balls. See The ball arsenal.

