It is pretty unlikely that you have broken your tailbone. Usually, fracturing the tailbone requires a hard blow to the area, for example from a fall or a car accident. In the absence of something like this, it is unlikely.
However, when doing sit-ups it is possible to 'bruise
' the tailbone or the surrounding structures. There is not much fat tissue over the tailbone to pad it. Therefore, if you are performing sit-ups on a hard surface like a concrete floor, this is a potential hazard.
Fortunately, the pain and swelling should heal up well on its own. Probably all that you need to do is rest the area (avoid sit-ups for now!). You may also want to take an anti inflammatory medication like ibuprofen or naprosyn for a few days to bring down the discomfort. Of course, you shouldn't do any more sit ups until the pain gets better!
In the future you might want to think about performing a different type of exercise to strengthen the abdominal muscles. At very least, you should consider only doing sit ups on a well padded surface!
If the pain does not get better, you should talk to your primary care doctor