2005 Cadillac Deville air not blowing

There is no relay, there is a control module(panel), a speed control processor, and the blower motor. You can check for power at the blower motor connector with a test light across the two terminals. If it lights, you need a blower motor. If it does not light, most likely the speed control processor is at fault. You can try rapping on the blower and see if you get operation out of it. If it comes back to working, there is no telling how long it will work.