The approval for the COVID-19 Vaccine was an accelerated process. This is because of an increased interest in the COVID-19 Vaccine and funding.
The FDA explains how the Emergency Use Authorizations process works in this link. The FDA explains the ongoing surveilance of the safety of the COVID-19 vaccines at this link.
Full FDA approval of the PfizerNBiotech COVID-19 vaccine is anticipated in early Fall 2021.