Most fighter kites are small single line kites that fight by manuvering to cross lines with another fighter and friction cut through the flying line. Once the tether is cut, the loosing kite floats free to the ground (hopefully). http://www.csun.edu/~ghsiung/fighters.html
will show the fighters of various countries.
Japanese rokkakus also have fighting tradions. It's more common around here to see a rokkaku battle. More about them here. http://www.gombergkites.com/FAQ-R.PDF
The battles are fun, good natured demos mostly. And there's more technics then just flying across someone's line. It's possilbe to crash a kite by tipping it over. Running, pushing and shoving isn't unheard of between friendly battlers. Cutting bridles and ripping sails are also effective technics, but a little more difficult to maintain friendships with.