A mutex is a lock -- if multiple users attempt to access a resource simultaneously, they each attempt to acquire the lock serially, and those not first are excluded (blocked) until the current holder releases the lock.
Mutexes are typically implemented with semaphores:
First a more developer-oriented explanation (because I imagine that's what you really want):
A semaphore locks a shared resource by implementing two operations, "up" and "down". First, the semaphore is initialized to some positive integer, which represents how many concurrent "users" (typically threads) can use that resource. Then, each time a user wants to take advantage of that resource, it "down"s the semaphore , which atomically subtracts one from the current value -- but the key is that the semaphore's value can _never_ be negative, so if the value is currently zero, "down"ing a semaphore implies blocking until it's non-zero again. "Up"ing is just what you'd imagine: atomically increment the semaphore value, and unblock someone waiting, if necessary.
Semaphores are generally seen as a fundamental primitive (one reason being that locks/mutexes can be implemented trivially as a semaphore initialized to one), but they also have broad use as a general synchronization mechanism between threads.
For a true ELI5 of semaphores (I enjoy writing these):
Imagine everyone in class needs to write something, but there are only three pencils available, so you devise a system (a system of mutual exclusion, per se) for everyone to follow while the pencils are used: First, you note how many pencils are available -- three. Then, each time someone asks you for one, (which we call "down"ing the semaphore) you subtract one from the number in your head and give them a pencil. But if that number is zero, you don't have any pencils left, so you ask the person to wait. When someone returns and is done with their pencil, you hand it off to someone waiting, or, if nobody is waiting, you add one to that number in your head and take the pencil back ("up"ing the semaphore). It's important that you decide to only handle one request at a time (atomicity) -- if you tried to do too many things at once, you might lose track of your pencil count and accidentally promise more pencils than you have.