Authenticator
An authenticator is something a person possesses or knows and uses to demonstrate control of a digital account during authentication.
More than an authenticator app
A password, a one-time-password generator, and a cryptographic security key are all authenticators. The term describes what you use to authenticate, not just the mobile app with that name.
An authentication factor is the kind of evidence it supplies. A password supplies knowledge. A cryptographic authenticator that requires a PIN can combine possession of a key with knowledge of the PIN.
A session cookie has a different job: it keeps an established session going. That is why you can stay signed in after putting away your security key. NIST calls this a session secret.
The account’s enrolled methods matter
Microsoft distinguishes which methods support sign-in, MFA, and recovery. A method listed on an account does not necessarily work for all three.
When replacing a lost authenticator, decide both who can enroll the replacement and what happens to the old method. Removing an authenticator and ending existing application sessions are separate questions. The recovery guide follows those decisions through the help desk.