Memorize this thing. It's very useful to know!
Not to get picky with my pal, b0b, but instead of 'memorizing' the circle, which really is just a map of some ordered information, I find it better to just understand the underlying principle, so that you can re-create it on your own any time you need to (rather than try to 'remember it' per se). So, the main use of this (at least as far as I know) is to tell you the 1, 4, and 5 chords in any given key. By now, I've memorized them, but before I had them down, I'd simply
count my way up the scale to them. So, in the key of C, the 4 chord would be the fourth letter up the C major scale (C,D,E,F: aha, it's 'F') and the 5 chord would be the next one (G). Done. You can do that for any key you need to know the 1, 4, 5 in (which would be for just about every song in western music, though you do have to know which letters to sharp or flat along the way).
Now, you can put all this in a circular map and you'll re-create the famous Circle of Fifths (or Fourths, if you head counterclockwise), but I find it much easier to just count up any scale to the 4th or 5th tone, instead of trying to memorize a map. Of course, that could just be me, so YMMV.