Battle:
Internet/Network: It doesn't get much simpler than this. All the players are in a largish arena, with no exit or enemies. Instead of shooting the enemies, they shoot eachother. If someone loses all of their five hearts, they die. Whoever deprived them of their final heart gets a point. The first person to reach a certain number of points (called roasts, and set by the server) wins the game, and the server moves on to the Next Level.
Before you play Battle, abandon any facts you have picked up about Weapon Damage from Single Player mode. All unpowered up guns (except for Ice and TNT) do one damage (that is, take away one heart). All powered up guns (except Ice, TNT, Pepper Spray and Electro Blaster) do two damage. Stomping, Sugar Rushes, Uppercutting and Sidekicking also do two damage each.
Local Splitscreen: This is just about the same. About the only difference is that the players decide on the various settings together. Also it can only use levels from the Home Cooked Levels list, so instead you should probably host a splitscreen network game. That way you can use any level, but the effect is the same, and nobody can join unless they're on your network.
Capture The Flag: (commonly known as CTF)
Internet/Network: Ok, this is a bit more complicated. By default, all the Even player numbers (Pressing F9 once will show you the player list) are on the Red team, and the Odd players are on the Blue team. You are able to set your team in the character setup, but please don't.
Now, in a (normal) CTF level, there are two bases (represented by Devan's Time Machine with Eva standing next to it), Red and Blue. If Team A's flag is at the A Base, and a member of Team B goes to the A Base, that member of Team B "captures" the A Flag. This means it floats behind him/her, and won't go away.
If the member of Team B holding the A Flag goes to the B Base, and the B Flag is at the B Base, Team B scores one point. The first team to reach a certain number of points (set by the server) wins the game, and the server moves on to the Next Level.
However, if the Team B guy with the A Flag goes to the B Base, but a member of Team A has captured the B Flag, Team B is unable to score.
So what happens? Neither team can deliver the other team's flag
until the player holding their flag is killed. For example:
RedGuy1 captures the Flag
RedGuy1: HAHAHA
BlueGuy1 captures the Flag
BlueGuy1: 0wned =D
RedGuy1: Now I can not deliver the flag.
BlueGuy1 has roasted RedGuy1
BlueGuy2: SCORE
(There's not too much point to saying this)
Blue Team has scored a point
Got it? Once RedGuy1 died, BlueGuy1 could deliver the Red Flag to the Blue Base. If, say, RedGuy2 had gotten to the Blue Base first, he/she would have captured the Blue Flag, and another roast would have to occur.
Understand so far? Good, as there's another feature.
While in a CTF game, you can use the feature known as Teamchat. This allows you to use the chat function, but only members of your team can read it. Well, assuming everyone joined as Default Team;
If you're an odd player number, odd players can read your Teamchat. If you're even, only even players can read it.
How does it work? Well, instead of pressing just T to start typing, hold down Shift while pressing T. Your "Chat:" header should become your team color. If not, you did it wrong.
Blue Team Teamchat will appear like this: "[BlueGuy3:]". Red Teamchat, due to a bug, doesn't get the brackets, so you'll have to assume it's Teamchat.
"I don't have time to Teamchat! I'm being hunted by the other team!" Well, fortunately, there is a fairly universal CTF Code which can quickly inform your team mates what's going on.
Click here for the CTF Code.
It's not as complicated as it sounds. You'll get used to it. Two more things; in CTF mode (not Battle), TNT can do one damage. Also, you only have a maximum of three hearts, not five.
Local Play Splitscreen:
Never play CTF using the normal Splitscreen. It just won't work. Instead, use the Network method described in Battle, at which point it works basically the same as the Internet version.
Treasure Hunt: (Occasionally referred to as TH)
Internet/Network: All the players are in a largish, non-linear arena, wherein are scattered lots of gems. There is also, in at least one place in the level, an Exit. The first person to reach the Exit wins the game, but it's not quite that simple.
Before you can Exit, you need to have a certain number of gems (set by the host, and usually 100). This brings up the problem of "What if nobody has enough gems and all the gems are collected"?
Well, while in Treasure Hunt you can not be killed at all, shooting people does have an effect. They lose gems. And depending on the strength of your attack, more or less gems are lost. A single blaster, for example, might make Person A lose three gems, while a Seeker Missile could take away ten.
"Lose" gems? Yes. When you lose gems, your gem count goes down, and that many Red gems (no matter what colors your original gems were) fly out of you, to be collected by whoever is nearby. Don't fight too much at first - go for the big gem spots before attacking others.
Local Play Splitscreen: Treasure Hunt works pretty well in Splitscreen, though it still has the HomeCooked Levels List problem, so you should use Splitscreen Network instead. There is no real gameplay difference.
Cooperative: (More commonly called "Coop" or "Co-op")
Internet/Network: This should probably be avoided. You can only play Coop mode like this with various external programs, and it doesn't work too well - only the server can be hurt, and stomping enemies will hurt the stomper. Plus it has problems with changing levels. Just avoid this.
Local Play Splitscreen: Here Cooperative mode works fairly well, though of course it uses the HCL list.
"But what is it?" Well, it's Single Player, only Multiple Players. Whatever number of people all play through whatever SP levels you like, and if one finishes the level, they all do. Each player has their own score and ammo levels, so there's no reason to let Player C have ALL the goodies.
Sadly, it's not perfect. Every player on a single computer has a few things the same, including Current Water Level and the Triggers. This is why sometimes if you join a server in Splitscreen mode, you may get kicked because of "Trigger problems".
Race:
Internet/Network: Just... don't. Like Coop, it requires a launcher. Race can be replicated to be Internet Friendly in Treasure Hunt Mode, but the same rules basically apply, so you needn't worry. Just don't try to use the real Race Mode.
Local Play Splitscreen: At the start of a level, none of the players can move. There is a countdown, then they all can move. At the end of the (normally linear) level, there is a warp which goes back to the start. But this warp has the added feature of adding one point to your score. The first person to reach a certain score (known as laps, and set ahead of time) wins, and the game moves on to the next level.
Remember, in order to win, you need to be fast. Make each lap (moving through the level on the way to the Final Warp) take as little time as possible. There will be lots of obstacles in the level, so get through them as quickly as possible.
Those are the five Multiplayer Modes. People have created other gametypes, but they are still done while in one of these five (generally Battle, CTF or TH). "Assassination" is a CTF piggy backer, while "Tests" are either TH or Cooperative, and "Last Rabbit Standing" uses Battle mode.