Adjacent means "next to" and degeneration refers to a breaking down process. It sounds like your mother had a spinal fusion (two bones held together to prevent motion).
Adjacent segment degeneration (ASD) is a common problem after spinal fusion. The vertebrae on either side of the fusion can become unstable. Increased motion at these levels causes even more break down of the nearby segments. Adjacent segment degeneration is the number one cause of spinal fusion failure.
This problem presents itself in many different ways. The vertebrae might slip forward closing off the space for the spinal nerves. This is called stenosis. The discs in between the vertebral bones can start to break down. The facet joints in the spine are damaged. The vertebral bones can even break.
Up to 50 percent of all patients with lumbar fusion show X-ray changes linked to ASD. Not everyone with X-ray signs of ASD have problems. They may still be able to function fully. But many patients end up having a second operation when pain and disability occur.