There are three types of dyes available:

Prismatic Dyes (Alchemy, Brewing, Pottery)
By far the most popular, prismatic dyes enable players to dye any visible armor slot, so that armor equipped in that slot will be tinted with that color. These dyes are made with tradeskills, but can be used without any skill. When you hear people buying or selling armor dyes, these are almost certainly what they mean.

Fine Plate Dyes (Alchemy, Make Poison, Pottery, for use in Smithing)
The very first set of dyes to be made available, fine plate dyes can only be used to tint fine plate armor. Fine plate is a line of basic, smithed plate armor. These dyes actually permanently change the color of each piece of armor. However, with the subsequent introduction of prismatic dyes, fine plate dyes have fallen into disuse. They are now useful mainly just by tradeskillers to improve skill in alchemy or make poison.

Ribbon Dyes (Brewing, for use in Tailoring)
These dyes are used to make ribbons (hence the name), which are, in turn, used to tailor robes. The robes can only be worn by casters (enchanters, magicians, necromancers, and wizards). Furthermore, the robes can only be made by the six races which can be one of those caster classes. While there is a market for some of the final robes, these are also useful primarily as a means of skilling up, particularly for tailoring.