I fell the other day and now my butt is swollen and I'm in extreme pain. Could I have broken somethi?

Butt pain. It depends on how you landed. Sacroiliac joint is commonly damaged when landing on the butt. There could me muscle contusion causing pain. Tailbone can also be damaged. A plain x-ray can reveal lot of information. See your primary care doctor for proper work up.
Fracture is painful. Extreme pain in the injured area of the butt, after one falls on his butt, is often a sign that some bone is broken. The tailbone, part of the pelvis bones, or the upper part of the thigh bones are all in that area and can be break with a fall. An orthopedic doctor or emergency room doctor can evaluate and do x-rays if needed, to get a diagnosis.