Harry's lightning bolt-shaped scar on his forehead becomes a major plot point in Harry Potter and the Goblet of Fire. The scar is the only mark on Harry's body from when Voldemort tried to kill him with an Avada Kedavra curse when he was a baby. It's also always been a way for strangers to identify Harry, whether he wants to be recognized or not.
But now, it's more than just proof of what happened to Harry thirteen years before – it's a tangible symbol of Harry's magical connection to Voldemort. The horrible pain that Harry feels when Voldemort casts the Cruciatus Curse is concentrated right on this scar. And his dreams of Voldemort's activities also cause his scar to ache. Professor Dumbledore puts forward a theory, and we trust his word:
It is my belief that your scar hurts both when Lord Voldemort is near you, and when he is feeling a particularly strong surge of hatred [...] Because you and he are connected by the curse that failed [...] That is no ordinary scar. (30.164-6)