They're used to get user input, for setting the clock perhaps, or for favorite stations, or a snooze button. Sometimes these are SMD buttons, which can't be removed from their board, making them mostly useless for our purposes, though the board may sometimes be removed as a whole and attached via ribbon cable to other projects. Often, though, they're standard through-hole mount buttons, which can be easily desoldered and reused.
In the clock itself, these are used for timing, and often quite small, with values in the picofarad to nanofarad range. The radio uses them for amplification, where they'll be larger, possibly electrolytic, and on the order of nanofarads to microfarads. The unit as a whole will have a few to keep power steady, and these can be pretty much any value from a few microfarads to a millifarad.