Does that mean the children would be elligible for adoption?

Do you mean to sign over the rights to the other parent? The answer would be no. Children are only eligible for adoption once the Court has terminated the parental rights.

Actually, I'm going to have to disagree here.

If someone is waiting to adopt the kids (through an arranged adoption, or a step-parent, etc), then signing over their parental rights is the first step in the adoption process.

Sorry I was so vague on the initial question. When I say sign over parental rights I mean, to another family member... it is my cousin. I just want to know what it means if she were to sign over the children to my husband and myself? Could we then proceed with adoption of the children?

In order for anyone to adopt a child, the biological parent(s) must sign a form giving their consent for the person in question to adopt the child. If one parent has custody and wants a "significant other" to adopt the child, thereby taking on the role of being the other parent, then the other biological parent must sign his/her consent, unless (s)he is deceased and the other parent can provide legal documentation (i.e. a death certificate) or has unknown whereabouts. In the case of unknown whereabouts, a judge would most likely require the custodial parent to make every good faith attempt to locate the other parent, in an effort to get him/her to sign the consent form, and document his/her attempts to do so. If, despite the custodial parent's best efforts the other parent cannot be located, then after a prescribed waiting period (say 90 or 180 days), the judge would authorize the adoption without the other parent actually signing consent. Now, if neither prospective adoptive parent has biological ties to the child in question, then both parents must sign consent before anyone else can adopt the child. An exception might be that, if the biological father has never been identified, the adoption may proceed with just the mother's consent.