POSIX allows mutexes to be recursive. That means the same thread can lock the same mutex twice and won't deadlock. Of course it also needs to unlock it twice, otherwise no other thread can obtain the mutex. Not all systems supporting pthreads also support recursive mutexes, but if they want to be POSIX conform, they have to.
Other APIs (more high level APIs) also usually offer mutexes, often called Locks. Some systems/languages (e.g. Cocoa Objective-C) offer both, recursive and non recursive mutexes. Some languages also only offer one or the other one. E.g. in Java mutexes are always recursive (the same thread may twice "synchronize" on the same object). Depending on what other thread functionality they offer, not having recursive mutexes might be no problem, as they can easily be written yourself (I already implemented recursive mutexes myself on the basis of more simple mutex/condition operations).
What I don't really understand: What are non-recursive mutexes good for? Why would I want to have a thread deadlock if it locks the same mutex twice? Even high level languages that could avoid that (e.g. testing if this will deadlock and throwing an exception if it does) usually don't do that. They will let the thread deadlock instead.
Is this only for cases, where I accidentally lock it twice and only unlock it once and in case of a recursive mutex, it would be harder to find the problem, so instead I have it deadlock immediately to see where the incorrect lock appears? But couldn't I do the same with having a lock counter returned when unlocking and in a situation, where I'm sure I released the last lock and the counter is not zero, I can throw an exception or log the problem? Or is there any other, more useful use-case of non recursive mutexes that I fail to see? Or is it maybe just performance, as a non-recursive mutex can be slightly faster than a recursive one? However, I tested this and the difference is really not that big.
The difference between a recursive and non-recursive mutex has to do with ownership. In the case of a recursive mutex, the kernel has to keep track of the thread who actually obtained the mutex the first time around so that it can detect the difference between recursion vs. a different thread that should block instead. As another answer pointed out, there is a question of the additional overhead of this both in terms of memory to store this context and also the cycles required for maintaining it.
However, there are other considerations at play here too.
Because the recursive mutex has a sense of ownership, the thread that grabs the mutex must be the same thread that releases the mutex. In the case of non-recursive mutexes, there is no sense of ownership and any thread can usually release the mutex no matter which thread originally took the mutex. In many cases, this type of "mutex" is really more of a semaphore action, where you are not necessarily using the mutex as an exclusion device but use it as synchronization or signaling device between two or more threads.
Another property that comes with a sense of ownership in a mutex is the ability to support priority inheritance. Because the kernel can track the thread owning the mutex and also the identity of all the blocker(s), in a priority threaded system it becomes possible to escalate the priority of the thread that currently owns the mutex to the priority of the highest priority thread that is currently blocking on the mutex. This inheritance prevents the problem of priority inversion that can occur in such cases. (Note that not all systems support priority inheritance on such mutexes, but it is another feature that becomes possible via the notion of ownership).
If you refer to classic VxWorks RTOS kernel, they define three mechanisms:
Again, this varies somewhat by platform - especially what they call these things, but this should be representative of the concepts and various mechanisms at play.