What is the circle of fifths for?
Going clockwise, each key is a fifth above the previous one and gains a sharp in its key signature; going counterclockwise, it gains a flat. Keys that sit next to each other share almost all their notes, so it is easy to modulate between them.
The inner circle gives the relative minor of each major key: same key signature, same notes, but a different starting note.
Finding the chords of a song
The three major chords of a key sit side by side on the circle: the key itself and its left and right neighbours. Their relative minors, just below, complete the most used chords.