My grandmother died; and left a house to my mother and I. Both our names are on the deed. I moved into the house, promising to keep up the house as well as live in it. I pay the utilities - now though, my mother is claiming it is her inheritance. My question is, if she proceeds to try and sell the house, is it legally hers, mine or ours to have to sell and receive the proceeds from? If she tries to sell without my consent, is that a breech of the deed?

If both names are on the deed, she can not sell the house without you signing for it.

It really depends on how the property is titled. The most common form of joint ownership is "joint tenants with rights of survivorship," often shortened to simply "joint tenants." Under this form of ownership you both own the entire property together in its entirety, so both must sign off on the sale. This is the ideally suited to married couples or partners in along term relationship. But if it's titled as "Tenants in Common" ("TIC" for short) then each person owns a specific share (perhaps 50% each, but not necessarily) and can sell his/her share as they desire without the other's consent. This is typical for unrelated people who don't plan to share residence in the property. In this type of ownership if your mother decided to sell on her own then she sells her half ownership to someone who then becomes a joint owner with you. Bottom line is: find out how the property is actually titled.

I'm curious why your mother thinks it's her inheritance if your name is on the deed? How did that come to pass? Did your grandmother gift the property to you and your mother while she was still alive? Or did she maintain ownership while alive and specify in her will that the property was to be split between you and your mother?