Well... I hesitate to give any advice on this, as my parents are at worst indifferent to my involvement in the fandom, but I do have some questions for you.
1. Are you financially dependent on your mother?
2. Does she dislike them because of the sexual association? What exactly does she dislike about them?
3. How comfortable would you be giving a slighted answer (e.g. just describing it as a cartoon)?
The definition of furry, from my point of view, is very loose. It can mean anything from someone who uses an animal as his or her online presence (a la Bearing or Atheism is Unstoppable) to the usual definition of someone with a sexual interest in anthropomorphic animals or an actual anthropomorphic animal. I myself am not comfortable giving half-true answers, but I don't know what your relationship with your mother is like.
If your mom is afraid of them the way people are afraid of clowns, then I'd guess the biggest danger you'd be in would be if you were a fursuiter. I'd expect you'd get a lot of nagging over it... my best advice is to find an understanding adult. I don't know where you live; most people in person I've talked to about this have never heard of furryism, and are themselves apathetic over the sexual involvement (I tell people that most members discover it through searching for fetish art, which may or may not be true). Your mother might not be willing to understand (or she might, again, I don't know), but there are definitely people out there who are.