Your name doesn't change automatically when you marry. The name you were born with or acquired through a previous marriage remains the same unless you take steps to change it.
If you are a woman wanting to take your husband's name, you simply start using it - consistently. A name change for a man may require a court order. Couples may also choose to hyphenate their surnames. Before you wed, ask the court clerk about any local requirements and how marriage certificates should be worded.
Change your name on important documents like your will and retirement plan and notify your doctor, dentist, lawyer, employer, insurance companies, credit card firms, bank and brokerage firm. You will also have to change your official identity with government offices like Social Security, tax agencies, the passport office, voter registration, the Department of Motor Vehicles, Veterans Administration and the Post Office.
In most cases a simple letter will do, but a certified copy of the official marriage certificate will be required by many government agencies.
Find name-change laws in your state at http://www.namechangelaw.com/states.htm