A semaphore with a count greater than 1
Suppose we installed the traditional key-based lock in the kitchen. The way this lock works is that if you have a key, you can unlock the door and go in. Anyone who uses this lock agrees that when they get inside, they will immediately lock the door from the inside so that anyone on the outside will always require a key.
Well, now it becomes a simple matter to control how many people we want in the kitchen — hang two keys outside the door! The kitchen is always locked. When someone wants to go into the kitchen, they see if there's a key hanging outside the door. If so, they take it with them, unlock the kitchen door, go inside, and use the key to lock the door.
Since the person going into the kitchen must have the key with them when they're in the kitchen, we're directly controlling the number of people allowed into the kitchen at any given point by limiting the number of keys available on the hook outside the door.
With threads, this is accomplished via a semaphore. A plain semaphore works just like a mutex — you either own the mutex, in which case you have access to the resource, or you don't, in which case you don't have access. The semaphore we just described with the kitchen is a counting semaphore — it keeps track of the count (by the number of keys available to the threads).