The first time I saw the word "modem" was (curiously enough) in a Sunday Tank McNamara strip in the late 1970s. A sports reporter had been sent to the boondocks to cover some local sports event. He was frantic about the lack of facilities at the stadium; he screamed "Where's my modem!!?!?!" in panic. One of the other 'casters held up a pigeon in a cage and said "Meet 'Modem' ".
When, in college, I got to actually use one of the thingies, modems were the strangest-looking things I'd ever seen. They were boxes about 9 inches long, 5 inches wide, and 3 inches high1, with these two round cups on top.

You guessed it, this was an acoustic modem. The modem would translate an acoustic signal, not an electric one. You dialed the number2, waited for the carrier signal to sound on the phone's handset, and then, you put the handset in the cups. You had to remember which cup to put which end of the handset in because it wouldn't work the other way.

My university had 300 baud modems and 1200 baud modems. The 300s would print (sometimes on a green screen, but often on HARDcopy, on a teletype), about as fast as you could type. The 1200s could print fast enough that you'd see a whole line appear at once! You could go down to the computing center and get on a hardwired 2400 baud terminal; that was really something. When I would complain about having to use a measly 300 baud modem instead of the snazzy 1200s, the old-timers would say "Consider youself lucky that you don't have to use a 110 baud modem anymore!"
120cm x 12cm x 5 cm.
2I remember holding the handset up to my TV speaker so that my VIC-20 could perform tone dialing, because that was faster than hand dialing on a rotary phone.