Master Core is summoned if you kill Master Hand as you VS both hands as @
CyberHyperPhoenix
has said.
Master Core is a creepy looking thing, it's first form is a plethora of blades that pretty much want to take your soul, it launches blade projectiles and swipes the blades like it's his hand. You also cannot see the amount of HP Master Core has on this form
Master Core, once beaten to a pulp, he'll transform into your player character, but it's a black shadowy figure/silhouette of your character AND it's also large (like a buffed up Super Mushroom)
Once THAT'S done, you beat the crap out of the core itself (it won't retaliate) and knock it off the stage