Nobodies are the body and soul left behind when a living being's heart becomes corrupted by darkness and turns into a Heartless. In effect, the name "Nobody" is a bit of a misnomer, as a Nobody is a body lacking a heart.
In the Kingdom Hearts series, all people that lose their hearts and become Heartless either by greed or death also have a separate entity created called a Nobody. The vast majority of Nobodies have their forms warped, bearing no resemblance to their former selves, but unlike Heartless, retain their memories, and their ability to think and plan. However, without hearts, they are incapable of feeling emotion, and are therefore ruthless enemies.
When people with unusually strong hearts have their hearts taken from them and become Heartless, the resultant Nobodies retain a human shape that is similar if not identical to their original appearance. Because they appear no different from a normal person, they are able to enter society and be treated as normal. The members of Organization XIII are examples of such Nobodies. However, these Nobodies are unable to survive if their opposites come back. For example, in the original Kingdom Hearts, Sora briefly lost and then regained his heart. At the moment he lost his heart, a Nobody was also spawned from him that would take the name Roxas. After Sora regained his heart, Roxas continued to live until the day he met Sora, at which point the Nobody merged with the original.
As stated before, Nobodies lack emotion because they have no hearts. The more powerful Nobodies however, like the ones in Organization XIII, put on a charade that they possess emotion using their memories of their emotions to fool themselves as well as others, demonstrating the behavior of sociopaths. It is hinted that some Nobodies do have emotions but this is because their Other(the person they were before is still alive) though this is usually unnatural since Nobodies are born when their Other loses their heart. Examples are Roxas, Naminé and possibly Axel.
Common classes of Nobodies introduced in Kingdom Hearts II:
The members of Organization XIII, first introduced in Kingdom Hearts: Chain of Memories and Kingdom Hearts II. The DS game Kingdom Hearts: 358/2 Days follows the story of Roxas's time as an Organization XIII member and thus allows the player to control the Organization members.
The truth about Nobodies (Kingdom Hearts 3D Spoilers)
It is revealed at the end of Sora's story in Kingdom Hearts 3D, that Nobodies actually can have hearts. Xemnas and Xigbar perpetrated the lie that Nobodies do not have hearts and convinced other members of Organization XIII that they needed to complete Kingdom Hearts in order to regain their hearts. Xemnas and Xigbar's actual plan was to hook up the members of Organization XIII to Kingdom Hearts in order to transfer the heart of Master Xehanort into the 13 individuals basically turning them into 13 clones of him. With Sora and Riku defeating Organization XIII in Kingdom Hearts II, this convinced Master Xehanort that most of the members of Organization XIII were not up to the task.