Basically there are 3 categories of chimneys: Pre-fabricated, masonry and hybrid.

There is a trick to tell what type of chimney you have:

Real bricks in the firebox usually mean you have a masonry system. The top of the chimney offers clues as well. If you can see a clay type square or oval pipe protruding out the top of the chimney stack you have a masonry fireplace and chimney system.

If you can see a metal pipe with a metal cap that fits the pipe perfectly, protruding out the top, chances are you have some kind of a prefab.

Look at the firebox. Does it have a metal box? If so you have a prefab. If it is all brick and mortar but with the metal pipe protruding out of the top you probably have a hybrid system.