The umbilicus is the transition point between the blood of the foetus and its mother, through the umbilical cord that connects to the placenta. At birth, the umbilical cord is clamped a few inches from the baby, and then severed. This several-centimetre protrusion is left to shrivel and fall off, which takes a couple of weeks.
At this stage, most babies are left with an ‘innie’ belly button, while a minority get an ‘outie.’ There is no scientific explanation for this – it’s probably all luck.
This is very different from an umbilical hernia, where a small portion of the abdominal contents protrudes through a defect in the abdominal wall under the umbilicus; this causes a lump that bulges when you cough. In babies, these mostly disappear after a couple of years, but in adults they may need an operation. However developing one as an adult has nothing to do with the shape of your navel.