An error occurs when there is a problem with the making of the coin, not in the die itself. There are three different types of errors, planchet errors, die errors, and mint striking errors. Some examples are: strike doubling, laminations, struck through, off center, clipped planchets, capped dies, grease filled dies, die adjustment strikes, and die clashes.
Varieties are mistakes in the die and many coins get minted that are exactly the same. Some examples of different types of varieties are: doubled dies and re-punched mint marks.
Many collectors today don't understand the difference between the two and will advertise a doubled die as an error coin and so on.