Compressors (and limiters) are devices that reduce the dynamic range of the signal.
(see the Q&As re dynamic range and dBs)

This is useful when the dynamic range is greater than the dynamic range of the sound system, or where you need to control the dynamic range of an instrument or vocalist so that it 'sits' better (i.e. can always be heard) in a mix. Compression is nearly always applied to pop and rock vocalists.

What actually happens is that you set a threshold above which compression occurs, and then set a compression ratio. The higher the ratio, the more the compression. The ratio expresses how much the output signal will increase for a given increase in the input.

Look at this graph -

If the threshold is set at +14dBm as here, and the compression ratio is 10:1, then if the input level increases by 10dBm, the output level will only increase by 1dBm. If the ratio is set to 1:1, there will be no compression as each 1dB increase in input gives 1dB increase at the output.

Some compressors use a system called 'soft-knee' compression, where the full compression ratio set is only applied when the signal exceeds the threshold by a considerable amount. Between the threshold level and this level the amount of compression applied is gradually increased. This gives a more natural sound and a certain amount of dynamics are preserved.