A hologram is essentially a three-dimensional photograph. It is a reproduction of the "interference pattern" of laser light produced by a beam split into two parts. One part is reflected from the object we are to produce a hologram of, and the other part, called the reference beam, goes indirectly to the film via a mirror and lens. These two beams interfere with each other, causing an interference pattern.
It's the interference pattern that the photographic film records, unlike a regular photograph, which records the amount of light from a specific point. Every point in a hologram catches light waves that travel from every point in the object. Wherever you look at a hologram you see exactly how light would have arrived at that point if you'd been looking at the real object. So, as you move your head around, the holographic image appears to change just as the image of a real object changes. That's why holograms appear to be three-dimensional.
Also, unlike the photograph, each point of the hologram contains all of the information from the object so the whole hologram can be reconstructed from a fragment.