Man is the only animal that laughs at what we call jokes, but other primates can laugh too. Chimpanzees and other great apes can be seen laughing apparently just because they feel happy, but there is also evidence that they do have a certain sense of humour. Apes have been seen laughing at simple comic situations such as someone falling over, as described in this article. It hasn't been proved that they are seeing the humour in the situation as a human might, but it seems quite likely - and bonobos, who resemble humans very closely in some ways, have been known to play tricks on each other and show amusement at the results.
Such humour may be quite basic, but quite a few humans never get much beyond it! Laughter is a very ancient mechanism, and the ability to produce something like our laughter - though not necesarily in response to humour - may exist in other animals too, such as dolphins.