A semaphore is a signaling mechanism used to coordinate between threads. Example: One thread is downloading files from the internet and another thread is analyzing the files. This is a classic producer/consumer scenario. The producer calls signal() on the semaphore when a file is downloaded. The consumer calls wait() on the same semaphore in order to be blocked until the signal indicates a file is ready. If the semaphore is already signaled when the consumer calls wait, the call does not block. Multiple threads can wait on a semaphore, but each signal will only unblock a single thread.
A counting semaphore keeps track of the number of signals. E.g. if the producer signals three times in a row, wait() can be called three times without blocking. A binary semaphore does not count but just have the "waiting" and "signalled" states.
A mutex (mutual exclusion lock) is a lock which is owned by a single thread. Only the thread which have acquired the lock can realease it again. Other threads which try to acquire the lock will be blocked until the current owner thread releases it. A mutex lock does not in itself lock anything - it is really just a flag. But code can check for ownership of a mutex lock to ensure that only one thread at a time can access some object or resource.
A monitor is a higher-level construct which uses an underlying mutex lock to ensure thread-safe access to some object. Unfortunately the word "monitor" is used in a few different meanings depending on context and platform and context, but in Java for example, a monitor is a mutex lock which is implicitly associated with an object, and which can be invoked with the synchronized keyword. The synchronized keyword can be applied to a class, method or block and ensures only one thread can execute the code at a time.