It's a very easy way to learn the key signatures by depicting them graphically. With C major at the 12 o'clock position, the Circle of Fifths starts with zero sharps or flats, also called accidentals. Adjacent keys are a perfect fifth from one another.
You only need to remember a few simple rules. First, going clockwise around the Circle of Fifths, each key is a perfect fifth higher than the previous one. Thus, G is a perfect fifth higher than C, D is a perfect fifth higher than G, and so on.
Similarly, going counterclockwise, each key is a perfect fifth lower than the last one. F is a perfect fifth lower than C, B-flat is a perfect fifth lower than F, etc.
This rule is especially easy to remember if you associate sharps with up and flats with down. It's natural (excuse the pun!) to do so because sharps raise notes (e.g., C-sharp is higher than C) while flats lower notes (C-flat is lower than C).
At the bottom of the Circle of Fifths are enharmonic keys. Enharmonic notes are different notes played by the same key on a piano. From a music theory perspective, C-sharp and D-flat are two different notes, with different functions depending on their harmonic context, but they sound identical on the piano since there's only one key to play both notes.
F-sharp major and G-flat major both have the same number of accidentals. Most composers will avoid writing C-sharp major with its seven sharps, and instead prefer to write the simpler D-flat major with only five flats.
After C-sharp major, the next key going clockwise along the Circle of Fifths is actually G-sharp major, but that would require a double sharp (F double sharp to be precise) in the key signature, which is avoided. Since G-sharp is enharmonic with A-flat, composers instead write A-flat major to make the music much more readable.
In theory, the Circle of Fifths is really a spiral, and it would be possible to keep going ad infinitum. In practice, it's simpler, being limited to seven sharps and flats.
An alternate way of depicting the keys is a Key Signature Chart, which you can also download from this site and print out.