The actual act of notifying the user was performed by a daemon called comsat (short for "communications satellite").
[1] Because the sudden, unexpected printing of a block of text on a tty can be annoying if it overwrites more useful information on the screen that can't be easily regenerated, biff is not used very much any more.
Some modern MTAs do not even support comsat (the server process which listens for reports of incoming mail) making biff useless.
The general idea of the incoming mail alert has remained very popular even as the original biff and comsat have been almost completely abandoned.
There are many biff replacements, several with similar names like xbiff, xlbiff, cwbiff, kbiff, gnubiff, wmbiff, imapbiff and xbuffy.