There have been no recorded cases of an animal-human cross species (or hybrid). They do happen, as things like a mule (a cross between a horse and a donkey) or a liger (a cross between a lion and a tiger) can be done. The animals that are breeding have to be very similar species and there are no species on the planet that are similar enough to us. It is possible that we once bred with other species of human when they existed, as they have found DNA in humans that may have come from other species of human.
To get a hybrid animal the parents have to be very similar, not only physically but also they have to be very evolutionarily related, and have a very similar DNA. There is not animal similar enough to cross with humans and produce a viable descendant. That is why there’s not hybrid of human and animals.
There are some hypothesis suggesting that thousands of years ago, Homo Sapiens (the human of nowadays) were able to cross with other different but very related species of men called Homo neanderthalensis (Neanderthal).
I have not heard or seen a human cross breeding with another animal but this is quite an intriguing question and am begining to think broadly about it. However, there have been few cases where different animal species apart from humans have been cross bred.
Alberto has it spot on here – there are no animals that are similar enough to cross with humans and we are the only species in our genus that is currently alive today. However, we do have traces of DNA that suggest we may have cross bred with other ancestors in the human family tree in the past such as Neanderthals.