In Massachusetts, a couple can apply for a license at any city or town hall, pay a fee (varies by town/city), and in 3 business days, the license will be issued. It will be valid for 60 days.
The officiant must view the license BEFORE conducting the ceremony (I like to receive it at the rehearsal, if possible.)
In Massachusetts, there is no requirement for witnesses to sign the license.
AFTER the ceremony, the officiant will complete, sign, and mail the license back to the town/city hall where it was issued. The town/city clerk will usually be able to provide (for a fee) a certified copy, which can be used for changing names and other legal purposes. With mail time and processing time at City/Town Hall, it may take a week to 10 days before a certified copy can be obtained.
Other states have different requirements. For the current requirements, I suggest you contact the city/town clerk or visit the municipality's website.