Firstly, a flash bracket gets the flash up higher off the camera. This helps to give a better light angle, avoid red-eye and to cast people's shadows lower behind them.
Often, the worst shadows are when you turn the camera 90 degrees and have the flash to the side. A rotate or flip bracket will allow you to keep the flash above the cameras even when you turn the camera. This is a big help if you are shooting people indoors where there are walls behind them.
There are two main types of bracket for this purpose. Flip and rotate. The flip brackets have a hinge on the flash arm, so when you turn the camera, you flip the flash arm around so that the flash is above the camera again.
The other type is rotate. With this set up, the flash stays stationary and the camera rotates on a little platform.
The are pros and cons to each.
Along with a bracket, you will also need a cord to attach the camera to the flash. If you use a dedicated flash, you will probably want to get a dedicated off-camera show cord, which will allow the flash to act just as if it was still attached to the camera (E-TTL metering etc).