A card is considered to be a counterpart of another card if it is intentionally made similar in appearance or effect.
"Lady of Faith" and "Doriado" are considered counterparts since they were intentionally made similar in appearance. The trend of making counterparts that are similar in appearance is more frequent in earlier booster sets. "Barrel Dragon" and "Blowback Dragon" are also considered counterparts since the latter's effect was based on the former's. However, counterparts in effect are sometimes less obvious or even unintentional.
There are enhanced counterparts, such as "Muka Muka" to "Enraged Muka Muka", "Man-Eater Bug" to "Nobleman-Eater Bug","Hane-Hane" to "Hade-Hane" , "Levia-Dragon - Daedalus" to "Ocean Dragon Lord - Neo-Daedalus".
Another kind of counterparts are the Dark counterparts, released in Phantom Darkness. These cards are dark versions of several popular monsters, such as "The Dark Creator" or "Dark Armed Dragon". All of them resemble older monsters in appearance, name, ATK and DEF, etc, with the exception that they are all DARK.