In Battletoads, the Dark Queen only has one special move: spinning until she turns into a tornado. How much damage is done to the player is based on how fast she is spinning; if she is about to turn into a tornado, minor damage is done, major damage will be done if she is completely spinning. If you happen to get close to her and don't use any moves, she will use her leg to kick you.
The Dark Queen usually appears between stages to mock and taunt the Battletoads, but she also possesses powerful magical abilities to fight back with. In the first game, her primary attack consists of turning into a tornado and attempting to hit the player. In Battletoads in Battlemaniacs, she instead teleports and attacks the players with her staff, and in Battletoads and Double Dragon, she is capable of transforming into a flame and moving accross the floor, remaining untouchable by doing so. She will also attack by emerging from the flame and throwing fireballs. Usually, fighting the Dark Queen isn't very challenging, especially compared to reaching her.
The Dark Queen does not appear as a boss in the Game Boy or arcade games. However, she appears in Stage 3 of the arcade, in which she watches the fight between the 'Toads and Scuzz's Robo-Rat, mocking and laughing at the 'Toads whenever they get hit and panicking when the Robo-Rat gets damaged.
In the Battletoads TV Pilot, the Dark Queen is after Princess Angelica for possession of the magic amulet she wears. She fights (and easily overwhelms) Zitz and Pimple by turning into a black tornado. At one point her minions are referred to as the "Beast Police."