For anyone that hasn't taken the sixty-odd hours needed to play through the games.
Heartless; The darkness in a heart that's become sort of a monster. Some kind of black shape with yellow eyes, usually.
Nobody; the empty body left behind when the heart leaves. Various shapes and colors.
Both are controlled by Organization 13, the baddies in Kingdom Hearts. The Organization is made up of thirteen Nobodies. When a Nobody's previous form had a strong heart, the body sort of keeps some of its personality and intelligence. So the Organization members are organized enough to want their hearts back. Demyx was one of the later members to join the organization.
Hearts are what we are, right? So when someone loses their heart, they technically have no feelings, morals, personality, etc. Some say that the Organization members put on fake personalities to antagonize the heroes of the series.
The plan; by killing lots and lots of people and harvesting their hearts, then collecting those hearts, we can create a world called Kingdom Hearts that is the center of all wisdom, power, blahblah. Which will let the Organization retrieve their own hearts.
If you find yourself completely fascinated or need more info, drop a post. Much easier than scanning through an entire
wiki. <3