time machine external HD

Yes, if you have a newer iMac with Thunderbolt/USB-C ports.

see > Back up your Mac with Time Machine - Apple Support

If you have any older iMac with USB 2.0 or 3.0 ports, then any USB 2.0 or 3.0 external hard drive will work just fine for Time Machine backups.

Yes it is best if you first use Disk Utility to Erase and Format it "GUID Partition Map" "Mac OS Extended (Journaled)" and name it Time Machine or something of that nature.

After that, you should get an automatic pop-up message asking if you want to use the drive for Time Machine.