What to do if I want to obtain legal custody of my niece because my brother does not want her and the state will not allow her mother to have her?

For you to get legal custody of your niece you need to get a written document signed by your brother drafted by an attorney allowing you custody of your niece subject to a court order. From what you have written you seem to have a good chance of getting custody of your niece.

If your brother would normally have custody, but he would rather that you have and care for the child, then his designation as you as the custodian of the child would help in your case for custody.  It also helps that you are a relative of the child.  If there is already a custody suit in place, you would need to jump into the custody suit.  If dad is agreement, then you need to state that in your motion.  As far as the eventual success of you versus grandmother, that decision will depend on any agreements that can be worked out and the judge.  The judge does not automatically award children to grandparents.  Instead, the court will use what is called a "best interest of the child" standard.  The court will look at both of your requests and decide who can provide the best environment for the child in the long run. The court will look to a variety of factors, like the housing situation of both applicants (does child have suitable accommodations), income of the parties, any specialized training by the parties, who has had the child historically, and the resources available to both parties (who live closer to the better school, who has better transportation to and from school, who has better access to extracurricular activities).  No one factor will control over the other.  A natural positive that you would have is that you are younger than the grandmother.  This means that you would be available to care for the child in the long run.  As an olive branch, you may want to represent to the court that you have no objection to facilitating visits between the child and her grandparents to encourage as much contact with family as possible.   The more factors that you can develop in your favor the better your chances will be at obtaining custody.