The Super Mario Wiki informs us that the floating eyes appeared before in Mario 64, which is odd, because we don't remember them from anything except our nightmares. The objective of the mini-game is to move the wiimote in circles, making the eyes dizzy as they eerily follow your every movement with their, um, selves.
"Oh God -- everyone's watching! It's just like being at the mall!"
Once they become dizzy, the eyes explode into a cloud of smoke -- only to be replaced by increasingly bigger ones. The player who pops the most eyes wins. And then the nightmare is finally over, right? Not so -- it's only just began.
You see, most mini-games end with the winner doing a celebratory dance, but in this case, the celebration is interrupted when several small eyes appear out of nowhere and move towards the players. The winner starts screaming and runs away in terror, leaving the utterly shocked and terrified loser to be surrounded by the peepers in a scene that looks like what we think was happening off camera in The Blair Witch Project.
Meanwhile, the remaining big eye simply stands in the background, watching you in silence. The image then lingers there for entirely too long (about seven or eight seconds) before the normal game resumes. Keep in mind that the average age of the Mario Party player is probably, like, seven. If you're playing Mario Party with a son or younger sibling and you come across this mini-game, good luck explaining to them why this is happening to their character as they weep inconsolably.
"It was time that you learned there is no God, anyway."