There are actually many kinds of Character Types but only a few that you can apply for straight out. Here's an explanation of the ones you can choose from, here:

Competitive

Competitive characters are what most characters are. They have power levels and techniques, and generally compete with each other for battle dominance. For that reason, they follow almost all of the rules we have, and are allowed to participate in just about every event we hold.

Non-competitive

Non-competitive characters get flexibility at the price of usability. They can do nearly anything with their characters, so long as it isn't a huge change to the structure of the room (unless you have host/cohost approval) but at the same time, they are not allowed to participate in plot events without prior approval, and competitive characters may not gain anything from RPing with them. You are also not allowed to have or role play non-competitive characters unless you have at least one competitive character that you RP on a regular basis.

Declared Villain

Only competitive characters can be declared villains. They get twice as much PL allowance as usual, and are allowed to bend the rules a little as far as a starting character is concerned. For instance, they are allowed one free tech that can get them out of any battle, and are allowed their own personal army of minions. On the other hand, they must be a major antagonist to a majority of the room during most of the time that they RP (though not necessarily all the time), and are, because of that, constantly under a much greater threat of being killed.

Official Teacher

Both kinds of characters (competitive and non-competitive) may be Official Teachers. They are allowed to teach multiple students their techniques for half the required length of a non-official teacher. They also gain their player the ability to review solos for training when the training characters are students of their teacher character. Each player is allowed only one Official Teacher character, though one teacher may retire so that another may become a teacher at any time.