My sister's name is on the savings account with my mother. If my mother dies does the count become hers? I am down as beneficiary?

It depends on how your sister's name is on the account. If her name is on the account "for convenience only" then she is not an owner of the account; she can only write checks and access the account for your mother's benefit. When your mother dies, the account will be distributed according to the terms of the account (if they are payable on death), or her Will if she has one, of by the laws of intestate succession if she has no Will. If your sister's name is on the account as joint tenant, then she is an owner of the account, and the account will become hers upon your mother's death. If you are named as a beneficiary (payable on death to you), then you will receive the account whenever the owner(s) die. If the owner is just your mother, you will receive it when your mother dies. If the owners are your mother and sister, you will receive it when both your mother and sister die (if it is not changed). The mechanisms of joint tenancy and payable on death, however, are contrary to each other. I have never seen that.