Bichon And His Back Is Arched Up Like A Cat His Back Legs Are Not Moving Right. What Is This?

Do you see signs of injury? It sounds as if he may have a spinal injury. If I were you I'd get him to a vet ASAP. I've had many dogs and only one acted like this and it was after it was accidentally stepped on. He's good now but I got him to a vet quick.
Good luck

While trauma is possible IVDD (intervertebral disc disease) is more common in small breed dogs that start exhibiting neurological signs.  This occurs when the discs that separate the vertebrae calcify then rupture and compress the spinal cord.
Arching the back is a sign of either back or abdominal pain.  In this case it is highly likely to be back pain since he is unable to move his back legs.  When a dog goes "down" in the hind end due to disc herniation it is an emergency.  If he has not feeling or is unable to move his legs at all surgical decompression is the best option.  Medical management can be appropriate when dogs are still ambulatory and are just ataxic (walking drunk).
Take him to a veterinarian immediately.  With this case prognosis can be affected by time between onset of signs and treatment.

Get the poor pup to the vet as soon as you can, it could be that he was hit somehow, and needs attention soon. Hope this helps, good luck to you.