Richard,

The U.S. has what is known as a "compulsory mechanical license" in its copyright law for "nondramatic musical compositions." This means that the owner of a copyrighted song is required by law to permit others to record that song. In return, those who record the song are required to

1) notify the author before distributing copies

and

2) pay a licensing fee set by the Copyright Royalty Judges

Most of the fees are administered by the Harry Fox Agency in the US, so in practice as a recording artist you usually deal with them rather than the composer.

You can see the section of the copyright code that covers this,

here

Note that this compulsory license applies only to recordings, not public performances. Public performances are controlled by a different process. Usually, composers WANT their works to be performed publicly, and want to get paid for that. So they join a Performance Rights Organization such as ASCAP or BMI. That organization then administers the performance fees, which are generally paid by the venue, not the performer. So the club where I sing a Bob Dylan song is paying ASCAP or BMI an estimated blanket fee annually, and in theory Bob Dylan is getting a piece of that.