You must go to your local Social Security Administration office with proof of citizenship, proof of your identity (usually photo ID), and your marriage license.
All you need to do is to apply for a new card. With the application, you will need to provide the documentation that proves the reason that you wish to change the name on your card.
You go onto their website and find the name change form. You fill it out and mail it in along with a copy of your marriage license. Then they will send you a new one.
