A Bride is a person totally in the power of a vampire. The term originates with the "Brides of Dracula" in Bram Stoker's novel, and doesn't reflect the actual gender of the person in question. In the world of Anita Blake, it is first used of Nicky, who becomes Anita's Bride when she rolls him to protect herself in Flirt.
Brides are basically meant to be disposable energy sources and devoted servants to their master. They can feel their master's mood, and are compelled to do anything in their power to make their master happy. They are unable to disobey a direct order by the master, no matter how large or trivial the matter, until the master explicitly counters the order. The master can draw energy from the Brides, even to death, and this transfer happens automatically if the master is badly injured or too hungry or otherwise down on energy. This process is always one-directional, unlike with an animal to call or a human servant.