Bassicaly, a nobody is the 'shell' of a heartless, its former self. Its like a vessel, it has no soul, no heart, no nothing. Although a nobody is indeed whats 'left over' when a heartless is formed, it won't neccesarily look like the person did before becoming a heartless.
It should be noticed that a nobody is formed everytime someone becomes a heartless, but they don't always appear human. Those with weaker souls become the basic Nobodies, these are those silver jumpsuit guys. The stronger ones, however, retain their form, and their consioucness. They also retain memories of being a human.
These stronger nobodies have the absolute belief that they do have a heart, that they are, essentially, real. They are stronger, smarter, quicker, and wittier than the other nobodies, and lead the lesser ones. Organization XIII is the collaboration of these stronger nobodies. You mention in your question 'how did they become nobodies' well, simply, people with a strong soul, who became heartless, left a nobody. Eventually, these strong nobodies banded together and created the Organization. They wern't a group of people before, they never knew eachother until forming the organization.
An example of this is Roxas. Roxas is actually Sora's nobody, because in the first one Sora did, temporarily, become a heartless, and as such a nobody was created. As Soras soul was incredibally strong, his Nobody was 'born' with consioucness, smartness, e.t.c. who was Roxas. However, as Sora wa sonly a heartless for a short time, Roxas has no memories of being a human. Therefore, he has no way of knowing he is actually a nobody. Because he was this 'type' of nobody, he was initiated into Organisation XIII (Roxas is the thirteenth member) however he left to pursue a goal, which is why Axel tries to bring him back.