It's not as Intimidating as You Might Think
Getting married is a happy event, but it does involve a lot of unglamorous practical details. Aside from planning the wedding itself, you'll have to invest some time in updating your various accounts and forms of ID to reflect your new married name. That includes your passport, an especially crucial document if you plan to travel outside of the country any time soon. Getting a passport re-issued in your married name takes a bit of time and documentation, but it's not difficult and may not even cost you anything.
If Your Existing Passport is Less Than a Year Old
If you've had your current passport for less than a year, the process couldn't be easier. Use the special form DS-5504 meant for this specific use. Fill out the form and send it in, along with your current passport and an original document—your marriage certificate, for example—verifying your change of name. You'll also need to submit a new passport photo along with the application. There's no cost involved in doing the name change this way, unless you need to get your passport in a hurry. Then you'll pay the usual $60 fee for expedited processing.
If You're Eligible for a Regular Renewal
If your passport is over a year old, you might be eligible to use the renewal process to carry out the name change. To do that, you'll need to fill out renewal form DS-82 and submit it along with your current passport, a passport photo, and an original legal document supporting your name change. In this case, you'll pay the regular $110 fee for your passport, plus the same $60 fee for expediting if you should need it.
Not everyone can use the renewal process, though. If your current passport was issued while you were a minor, or if it was issued more than 15 years ago, or if it's missing or badly damaged, the State Department will require you to complete a full regular application.
If You're not Eligible for a Regular Renewal
If any of the above conditions apply to you, you'll need to complete a DS-11, the standard passport application form. That requires some additional documentation: evidence of U.S. citizenship such as a birth certificate, and a valid, current, government-issued ID. You'll also need to provide photocopies of those documents, and—again—a passport photo and an original legal document such as a marriage certificate to support your request for a name change. In addition to the standard $110 fee for your passport application, you'll pay a $25 "execution fee" to have your passport issued, and the $60 fee for expedited processing if you request that.
It Works the Other Way, Too
A name change isn't necessarily always because of marriage, and the same basic rules apply if you're changing your name for other reasons. If you're reverting to your birth name, for example, you'd use the appropriate application form but support your application with a copy of your divorce decree or a court order authorizing the name change. If you routinely use a different name for professional purposes or other reasons, you can even have a passport issued in that name as long as you can demonstrate that you've used it consistently for five years or more.