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 "Dryad" 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.
Several cards referred to as retrained counterparts exist, such as "Celtic Guardian" to "Obnoxious Celtic Guard".
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 basically the dark versions of several popular monsters, such as "The Dark Creator" or "Dark Armed Dragon". They share similar characteristics; all of them are DARK attribute and, as previously mentioned, they resemble older monsters in appearance, name, etc.
Another kind of counterparts are the Malefic cards. Not all of the Malefic cards are counterparts, but the majority of them, such as Malefic Blue-Eyes White Dragon, are.