A claw and ball foot is exactly what it says -- with the context that this is a feature of many fancy furniture designs. In the early to mid 1700s people were starting to experiment with imported design elements; the claw and ball foot was supposed to be a Chinese allusion, with the foot being a dragon's foot clutching a pearl (but not colored like a peal, and not the size of a pearl; it's like, representational, man). They are often paired with cabriole legs, to reinforce the appearance of a living animal. While Chinese dragons are often portrayed holding a large pearl, the idea that the proper place for a dragon is holding four of these priceless gems while hiding under a highboy is apparently a western invention.
The claw and ball is widely associated with Chippendale furniture, although it is certainly not unique to them. Around the end of the 1700s the claw and ball fell out of fashion, but not for long. In later appearances the English would often substitute a lion's claw for the dragon's, and in American furniture it's not uncommon to see it appear as an eagle's claw.