EDIT: As of January 1st, 2016, the US no longer adds additional pages to its passports. However, when you apply for a passport you may choose to receive a 52-page passport rather than the standard 28-page passport. Applicants outside the US automatically receive the 52-page passport. For more information, see the State Department's press release.
If your US passport isn't set to expire within the year, you can go to any US embassy that offers citizenship services and ask them to insert additional pages. You generally don't need to make an appointment for it, and they can typically do it on the same day, but it's good to call ahead to just make certain. There is also a small fee, so you should be certain to bring that with you in cash. Depending on the country, payment by credit card, local currency, or US dollars may be possible. If you are unsure, contact citizenship services ahead of time.
The actual insert is bound by two sort of security grade adhesive strips that cling to two pages in your passport, the first looks like this:
The second is a clear strip, similar to packing tape. You get a total of 12 extra pages. It's like taking the cover and spine off a small book, and putting it into a bigger one. For an additional fee, a maximum of two additional sets of pages (total 24 pages) can be added in a single visit.
So, if it looks like you're going to run out of pages, get this done while you still have a little room or the insert itself is going to need to go over top of stamps you'd probably want as keepsakes later. Many countries will refuse to issue entry permission if there aren't sufficient pages left in your passport (some countries requiring up to 6 blank pages), so be sure to check the regulations of any countries you will visit to ensure that you have enough pages to enter the destination country, and re-enter your country of residence.
If your passport is set to expire within (or close to within) the year, just renew it instead, the consulate at the embassy will probably ask you to do so anyway when you visit.