Why is my car making a loud grinding noise in the rear and is swaying back and forth when I'm driving?

If it is rear-drive, suspect a bad universal joint, differential, or transmission connection. If it is front drive, with a rear axle brake, this could be broken or mis-engaging. Swaying seems to indicate a faulty connection with the rear axle. It could actually have been broken at one wheel connection, but very shortly the wheel would have fallen off.