For a mixed-breed dog:
* Crossbreed
* Mutt
* Heinz 57 (informal)
* Mixed breed
* Hybrid
* Cur (archaic, derogatory)
For something of mixed origin or quality:
* Hybrid
* Mischling (German)
* Bastard (archaic, derogatory)
* Hybrid
* Mongrel (can also be used in this context)
For a person of mixed heritage:
* Mulatto (archaic, derogatory)
* Mixed-race
* Biracial
* Multiracial
It's important to be mindful of the connotations of each word. "Cur" and "bastard" are considered derogatory terms and should be avoided.