I twisted my knee and now can't move it. Is it broken?

Need x-rays to know. Twisting of the knee most commonly results in a ligament sprain and/or a cartilage tear. However, fractures and dislocations can result from a twist as well. Unless you improve very rapidly, i recommend a physician evaluation, and most likely, x-rays will be the initial imaging study. If your symptoms are severe, then emergency room evaluation is recommended.
Probably not. The tightness in the joint is more likely due to swelling deep inside the joint. It is possible that you have simply sprained or strained the knee and with rest, ice, and anti-inflammatory medications (motrin, aleve, (naproxen) etc) it will impove on its own. If not, or for severe, worsening pain, don't wait - see a doctor.