Uruk = Orc, hai = people or folk.
Sauron bred the original Uruk-hai. The ones that attacked at Amon Hen were of Saruman's special breeding. It's said in the books that he bred men and Orcs but he also added some resistances and abilities so maybe there was some magic involved. Saruman had a bunch of different crossbreeds running around.
The Silmarillion goes into detail about how Orcs used to be Elves. I won't go into a lot of detail but basically they were tortured and disfigured until they were made into the Orcs we see now. They still breed the same way, so why is it such an impossible or disgusting thought that the Uruk-hai could come about as a hybrid?