Mutual Exclusion. Here's the Wikipedia entry on it.
The point of a mutex is to synchronize two threads. When you have two threads attempting to access a single resource, the general pattern is to have the first block of code attempting access to set the mutex before entering the code. When the second code block attempts access, it sees that the mutex is set and waits until the first block of code is complete (and unsets the mutex), then continues.
Specific details of how this is accomplished obviously varies greatly by programming language.