By "dirty" is meant the type of oil product that these ships can transport. "Clean" is considered stuff like gasoline, naphtha and kerosine, while dirty products are usually things like gasoil and all sorts of fuel oils. So a dirty tanker can carry dirty products and crude oil while a clean tanker (that will need some coating first because of the more corrosive nature of clean products) will carry clean products, but after a proper tank cleaning will also be able to carry dirty products or crude.
This is a very general reply to give you an idea. Further reading could probably answer more of your questions.