Most beings are born as Complete Beings, with a Heart, Body, and Soul, and most beings live their lives this way without changing until their death when their Heart and Soul leave their Body behind. However, since the outbreak of Heartless on Radiant Garden after the studies of Ansem the Wise, it has become more common for Hearts to be separated from the Body and Soul, creating Heartless and Nobodies.
For more information on the composition of a Complete Being and their workings, see here: Life
It is possible to be born as a Nobody, without a Heart. If the pregnant mother uses Corridors of Darkness, or comes into contact with Darkness through other means, too often, her developing baby's Heart can be consumed by it and lost. Also, if the pregnancy is sped up through unnatural means, it can result in the baby's Heart never developing or being underdeveloped and lost quickly. The resulting child is born a human-form Nobody, with all the same abilities that they develop over time.
For more information on Nobody birth, see here: A Theory on Nobodies and Birth