Please refer to my first post on the Circle of 5ths to explain the basics.
So we've looked at what the circle means at a glance. Now let's get a little more in depth.
As you can see from the circle, there is a pattern to the sharps and flats in a key signature. The sharps increase one at a time in this order F C G D A E B. There are a few different rhymes people use to remember this, but the one I always learnt was: Father Christmas Goes Down All Escalators Backwards. The flat keys go in reverse, B E A D G C F (which is the rhyme backwards). Using this you can work out easily that if a key has 4 sharps, they will be F C G D. If a key is 5 flats they will be B E A D G.
Now as I said before, each key signature is shared by two types of scale - the relative major and minor. These two scales are always one and a half steps apart from each other (three semitones). To find the relative minor of a major scale, step 1½ steps down from the major. To find the relative major of a minor scale, step 1½ steps up. Have a look back at the picture of the circle, and see how that all fits together.
Now you should be able to work out what sharps or flats are in a particular key, as well as the relative major/minor of a key.