The "beachball" is the colloquial term for the busy mouse cursor displayed by Mac OS X when an application stops responding. In reality, the animated cursor does not depict a spinning beach ball but in fact shows a spinning optical disc with a multi-colored rainbow effect caused by refraction; nevertheless the term "beachball" is extremely popular among users, perhaps because of its ironic connotations (a beachball is a cheap, plastic toy without much functional merit).
The handling of beachballing has improved considerably in Mac OS X over the years. For one thing performance has improved in general, which makes the beachball appear less often. For another, while the beachball originally kicked in whenever the frontmost application stopped responding, Apple changed this (in Panther, I think, although I may be wrong) so that the beachball would only appear if the mouse cursor was hovering over a visible portion of the window of an affected application.
This was a great improvement because it meant that you were less likely to sit staring at the screen waiting for the beachball to go away because you didn’t realize which applications were affected and which weren’t. It also turned the beachball into a helpful diagnostic tool because it meant that you could check the status of all visible applications just by moving the mouse over them.