A modem takes an analog signal and converts it to a digital signal, and
vice versa.
No, not exactly.
So a modem is an analog to digital converter. So why is it
called a modem (modulator/demodulator)?
A modem is a device that takes a digital signal and converts it
into something that can be transferred over an incompatible
medium.
Mostly: transmit digital data over voice grade telephone lines.
A true analog to digital converter can convert any analog signal
into its digital representation.
A true digital to analog converter can convert any digital value
or stream into an analog signal.
A modem can only convert a digital datastream into some type of
'sound' that is compatible with the telephone system.
And a modem can retreive the same datastream from the received
'sound'.
The way a modem accomplishes this is by modulating sound (or
noise) in such a way that the receiving modem can reconstruct the
digital datastream.
And this can be done from very basic (one note for digital high
and another note for digital zero, as in 300 Baud systems) to
extremely complex systems, like V90 or V92.