How is magnetism used in a computer?

Magnetism is used to operate the fan motors and the disk drive motors as well as operating the "voice coil" that causes the disk read/write heads to move. The surfaces of the hard disk "media" is magnetized to represent data. The hard disk read/write heads are nothing more than small electromagnets and as the heads move across the spinning disk media, magnetic data is sent or received by the disk heads.
Older computers used a magnetic "core" to store data. That's the origin of the term, "core memory". With the decrease in cost ind incredible speed of transistor based memory, nobody uses magnetic core any more.

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