There is no specific form used to order passport applications.
The National Archives has passport applications from October 1795 - March 31, 1925; the US Department of State has passport applications from April 1925 to the present. We only have the indexes, however, through 1923. For records after 1925 (or for indexes after 1923), please see the information below about State Department records.
For a complete article on passport applications in the National Archives, please see:
The State Department Passport Services has passport applications from 1925 to the present. It also has the indexes for 1923-1925.
Passport Information from the State Department
Many of the passport records held by NARA have been digitized by
FamilySearch.org, Ancestry.com and Fold3.
See the entire list of digitized documents here (search for the term "passport" on the page):
For instance, the link to find the passport records on Familysearch.org are here:
Familysearch.org is a free website. Ancestry and Fold3 are both subscription websites, but there is free access from computers at any National Archives facility. Many public libraries also provide free access.