What is the difference in between "race" and "breed"? Could I say "human breed" instead of "human race"?
At the simplest level, it's race for people, and breed for animals.
From Oxford Dictionaries - race (emphasis mine): "Each of the major divisions of humankind". [It then goes on to give lots of specific meanings, just in case anyone thinks that's too simplistic a definition of a highly-charged word.]
And from the same dictionary - breed (emphasis mine again): "A stock of animals or plants within a species having a distinctive appearance and typically having been developed by deliberate selection."
However, the same entry then gives a second definition for breed: "A sort or kind of person or thing". So you could say, for example, that people who spend Saturday night posting to Internet forums are a special breed of people. That's a metaphorical usage - it doesn't imply that the people share common DNA.
And no, you'd never say human breed; the stock expression is human race.