Mutex: exclusive-member access to a resource
Semaphore: n-member access to a resource
That is, a mutex can be used to syncronize access to a counter, file, database, etc.
A sempahore can do the same thing but supports a fixed number of simultaneous callers. For example, I can wrap my database calls in a semaphore(3) so that my multithreaded app will hit the database with at most 3 simultaneous connections. All attempts will block until one of the three slots opens up. They make things like doing naive throttling really, really easy.