In UK the position is slightly more complex than either terence2 or alan suggest.

There are two sets of rights recognised in relation to the use of copyright recorded music:

Performing Right Society (PRS)
licenses the performing right in copyright music on behalf of composers and music publishers; and

Phonographic Performance Ltd (PPL)
licenses the public performance, broadcasting and cable programme rights in sound recordings on behalf of record companies and performers.

Strictly speaking, if you compile your own collections on CD or otherwise, you may also need a license from the Mechanical Copyright Protection Society (MCPS), which licences the making of mechanical copies (i.e. recording onto a CD, tape, video or DVD) of copyright music on behalf of composers and music publishers.

A complex subject.