Basically, both are fixed-equipment PVP modes involving shield bumps as your main weapon.

Shieldball: Soccer with shield-bumps. There are goals for each team, and the goal is to knock a ball into the enemy's goal using shieldbumps. Teams of 4-6 each, and costume gear is allowed, although anyone in MSI gear (Mercurial, BKC, Mercurial Demo, and whatever I missed) get their MSI deactivated. Also, everyone gets team-colored shields, because apparently, Proto Shields are no good at shieldbumping. Oh and you can't kill each other. Team with most points when time runs out wins.

Dodgeslime: Replace the ball with five to ten REALLY ANGRY melee-attack-spamming Jelly Cubes, reduce everybody to one pip of HP, and get rid of the goals, your objective being to kill the enemy Knights by knocking the jellies into them, or them into the Jellies. There's Random Team and Guild Team, where the jelly CAN KILL your teammates if you mess up, making it practice for avoiding shield-bump accidents in the Clockworks. And there's Free-for-All. No points guessing what that means. >:D Teams of 4-6 in Team matches, and maximum of 10 players in FFA. Costumes shown and no MSI, same as Shieldball.

Oh, and before I forget, Dash is allowed in both modes, and a non-damaging, non-stunning Shield Bash is allowed in Dodgeslime.