Separation anxiety is normal in very young children, 8-14 months old. Kids often go through a phase when they are "clingy" and afraid of unfamiliar people and places. But when a child over 6 has this fear, it's excessive, and lasts longer than 4 weeks, they may have separation anxiety disorder.
Some children also have physical symptoms, such as headaches or stomachaches, at the thought of being separated. The fear of separation causes great distress and may interfere with their normal activities, such as going to school or playing with other kids.