A mutex is a mutual exclusion object, similar to a semaphore but that only allows one locker at a time and whose ownership restrictions may be more stringent than a semaphore.
It can be thought of as equivalent to a normal counting semaphore (with a count of one) and the requirement that it can only be released by the same thread that locked it(a).
A semaphore, on the other hand, has an arbitrary count and can be locked by that many lockers concurrently. And it may not have a requirement that it be released by the same thread that claimed it (but, if not, you have to carefully track who currently has responsibility for it, much like allocated memory).
So, if you have a number of instances of a resource (say three tape drives), you could use a semaphore with a count of 3. Note that this doesn't tell you which of those tape drives you have, just that you have a certain number.
Also with semaphores, it's possible for a single locker to lock multiple instances of a resource, such as for a tape-to-tape copy. If you have one resource (say a memory location that you don't want to corrupt), a mutex is more suitable.
Equivalent operations are:
Counting semaphore Mutual exclusion semaphore
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Claim/decrease (P) Lock
Release/increase (V) Unlock
Aside: in case you've ever wondered at the bizarre letters used for claiming and releasing semaphores, it's because the inventor was Dutch. Probeer te verlagen means to try and decrease while verhogen means to increase.
(a) ... or it can be thought of as something totally distinct from a semaphore, which may be safer given their almost-always-different uses.