Dark Counterpart monsters are basically retrained, alternate dimension versions of familiar Yu-Gi-Oh! monsters. The TCG has a few official ones (for example: Dark Creator and Dark Armed Dragon). And we have created some more.
Dark Counterpart monsters need not be acually of a DARK attribute (but usally are). Sometimes, they require that a certain amount of DARK attribute monsters are in the Graveyard before you can Special Summon them. Others have effects which only activate when there are DARK attribute monsters in the Graveyard. Others may have effects which are activated by removing DARK attribute monsters from the Graveyard.