I bent over and felt muscles clench up in the lower center of my back. Now its throbbing and I'm having a hard time moving around. What do I do?

Osteopath. Sounds like muscle spasm or acute sprain of sacoiliac joint. Heat, rest, examination, and a manipulation might be all that you need. Breaking something inside your back by merely bending over is not really possible.
If the pain is only. In your muscles, rest and Ibuprofen should make it feel better. You need to make sure that you don't hurt it again, by using proper bending technics and exercise. But if the pain goes to legs or you get numbness or weakness of your legs, it could mean that nerves are affected. You will then need a doctor to advise you on what to do.