Semaphore and Mutex both are interprocess lock mechanism for C#. You can aquire a lock in one process and no other process can take that lock until released.

Mutex is a single lock and if any one thread has taken it no other thread in any other process or same process can take it until released.
static Mutex mutex = new Mutex();**

Whereas Semaphore is a mutex with a specified number of locks. You can specify the the number of lock can be aquired.
static Semaphore _pool = new Semaphore(0, 4);

The only thing I can add to the excellent answer above is that the best way to avoid deadlocks is not to have shared state between threads