The Move-Relearner is a blue-skinned NPC found inside Cerulean City's Pokécenter. He can re-teach your Pokemon moves they have learnt before. This can be useful for creating ideal movesets.
A 10,000 Pokédollar fee applies for each individual move re-learnt from the Move-Relearner. For instance, teaching 4 moves would cost 40,000 Pokédollars altogether.
The Move-Relearner's appearance was most likely based on Bebee2, a distinguished moderator of Project: Pokemon.