A card is considered to be a counterpart of another card if it is intentionally made similar in appearance or effect.
Early sets (especially from Vol.1 to Booster 7) recycled monster designs very frequently. The monsters were typically identical in design, different only in colors, names and stats. They are otherwise considered unrelated entities given their dissimilar names and the lore stated in their respective flavor text, with few exceptions.
"Barrel Dragon" and "Blowback Dragon" are 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", "Milus Radiant" to "Missus Radiant", etc.
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.