My situation is a little different.
My brother is 48 years old and has a 7 year old daughter. Her mothers rights were terminated when his daughter was an infant due to her health issues which has now left her on life support. Sad situation. However my brother is a wonderful father and of course we as his family have stood behind him and helped him along the way.
The problem now is he was diagnosed with terminal cancer back in June. He has since moved in with our mother which is 76. He has gotten to the point he can no longer care for his daughter. He asked me to take her and finish raising her.
So what do we need to do at this point? Will this be an expensive lengthy process?
And also if he passes before it is done what will happen to my niece?
All advice is greatly appreciated.
God Bless You All!

At this point the best plan would be for you and your brother to go to court and have you ordered legal guardian. This will grant you all the rights of a parent to make decisions/etc without removing any of his.

Since he has sole custody, he should include in his will his desire to have her remain with you. Upon his passing, you simply apply to have the guardianship transferred to full custody. If for some reason he passes before you can get to court, his desires stated in his will should still be enacted. The will can be done on your computer and notorized at most banks before the end of the day.

These are neither expensive or lenghty. There may be some local filing fees, but since everything is unopposed, its simply getting the court to rubber stamp your and your brother's decisions.

I'd add one thing to Steve's answer. On Monday, you and your brother should have a Power of Attorney signed that conveys on you the right to make decisions regarding the welfare of your niece. You can find POA forms on the Net (ex: Free Durable Power of Attorney Form) and need to have it signed and notarized. It would be better to have an attorney prepare it, but you can use a form. In the form I linked to I would change it to say:

1. Authority to Act. The Agent is authorized to act for me under this Power of Attorney and shall exercise all powers in the best interests of my daughter <insert daughter's full name>.
2. Powers of Agent. The Agent shall have the full power and authority to manage the care of <insert daughter's name> and to exercise my legal rights and powers with respect to her care, including those rights and powers that I may acquire in the future, including the following:

You can delete the sub paragraphs of 2. since they don't apply.

3. Durability. This durable Power of Attorney shall be irrevocable until the trust corpus is surrendered by the trustees, shall not be affected by my death or disability except as provided by law, and shall continue in effect after the surrender of the trust corpus until my death or until revoked by me in writing. Upon my death guardianship of <insert daughter's name> will be fully invested in the Agent.

I believe this can protect you in case guardianship proceedings cannot be completed prior to your brother's death.