Form base authentication is based on an HTML form where the user type in username and password. This is probably the more used way to authenticate users with APM.
Cert based authentication requires an user certificate to be sent as part of the SSL handshake as the APM will demand it. This authentication method is usually the chosen one when higher level of security is required (a certificate is not something easy to get).
It is usual to see a combination of both so the APM requires a certificate at first step and later on will show a form for the user to provide its credentials. This adds extra security.