Gather the necessary proof of citizenship. You'll need two copies of your birth certificate; one must be certified. Also bring another form of government ID and a photocopy of it.
Obtain three official passport photos. The photos must be two inches by two inches on a plain white background. The photos must clearly show your face; do not wear a hat or other items that may obstruct the view of your face.
Go to the Honduran consulate in person and complete the necessary paperwork. Form DS-11 is the main form you'll need to fill out. It is available for download and may be mailed in ahead of time, though you'll still have to visit the consulate to complete your application. Include your proof of identity/citizenship and passport photos with your application.
Pay the application fee in cash when applying. (Confirm the current prices with the consulate before your appointment.) It may take several days, or as long as a few weeks, to receive your passport.