On what principle does a tape recorder work?

A tape recorder, tape deck, reel-to-reel tape deck, cassette deck or tape machine is an audio storage device that records and plays back sound using magnetic tape, either wound on a reel or in a cassette, for storage. It records a fluctuating signal by moving the tape across a tape head that polarizes the magnetic domains in the tape in proportion to the audio signal.

https://en.wikipedia.org/wiki/Tape_recorder

The principle is that of electromagnetism.

http://www.school-for-champions.com/SCIENCE/electromagnetism.htm

In recording mode, the sound signal is converted into a varying electrical current which passes through the tape head. The tape head is an electromagnet i.e. the current turns the tape head into a magnet, the strength of the magnet depending on the strength of the electric current. The magnetised tape head then induces magnetism in the tape as the tape moves past the head, much as you can convert a steel nail into a magnet by stroking it with a permanent magnet. In effect the tape contains many millions of these sub-microscopic "nails", referred to as "magnetic domains" in star_gazer's answer.

In playback mode the tape is driven past the head which is not supplied with any electrical current. Instead, the moving magnets in the tape induce a weak current in the tape head and the current is the exact replica of the original current that created the recording. This current is gently led away then amplified and comes out of the speakers as "Africa" by Perpetuum Jazzile.

https://www.youtube.com/watch?v=yjbpwlqp5Qw