Try sorting servers by Player count to see the populated ones. The more populated ones generally provide something to do.
TDM - Team Deathmatch. Teams compete to kill each other and score objectives.
CTF - Capture the Flag. A Team Deathmatch where teams try to steal an item from the enemy's base and bring it to their own for points or round wins.
TF2 - A recently popular game type based off Team Fortress 2. You'll generally be able to pick a 'class' of weapons to use (for instance, being able to wield a Flamethrower or a Minigun, but not both) and then fight.
Dogfight - A deathmatch based around using planes and flying vehicles to travel between distant areas. Aerial combat and bombing runs to be expected.
CityRPG - A game mode where people do things like earn money, build a city and interact with each other. Role playing.
Zombies - Players try to hold off AI or player-controlled zombies that attack in hordes and break down walls.
Fort Wars - Players try and build fortresses (typically being given the Wrench to add interesting mechanics and Wrench Events) to hold off others or work in teams.
Challenge - Players, working alone, complete various challenges based off jumping between platforms, using special items like the Hookshot or Portal Gun, or player types like the Agile Player that can climb walls to reach exits or objectives. Rewards follow on completion.
Freebuild - Build anything you want! It's suggested not to build any projects or huge things that you will want to work on later, because you never know if it'll be gone.
Build - Servers that don't involve minigames or anything, only building. Minigames can occur though.
