Can anyone tell me how a HUD works?

I heared that a HUD consists of a mirror that projects a ‘ghost’ image onto the windscreen. Is that true?

Also, in most cockpits, I noticed that the HUD is mounted lower than the pilot’s eye level. Does that mean the pilot has to look down at the head-up display?

Could the mirror of a HUD be used to adjust the height of the ‘ghost’ image to match the pilot’s eye level?