Nebelah: An animal that has died a natural death or was killed not in
accordance with the jewish ritual lawnatural death is called "nebelah"
(carcas). (Bechoros, 3a)
Terefah: An animal afflicted with an organic disease or disability
(eg. the removal of a certain portion of the knee) (Chulin, 42a)
A first-born-Nebelah can have kedushat-Bechor and needs to be redeemed from a Kohen. If it is terefah it loses its Bechor-status and doesn't need redeeming.
According to Rav Hisda, if, when it died, a non-Jew owned any of the parts of the Nebelah that would render the animal treif if that part was missing (eg. windpipe), then the animal is unfit to be a Bechor, and doesn't need to be redeemed. (Bechoros, 3a)