Paris is the "wanton" (i.e. horny) Trojan prince who started the Trojan War when he stole Helen from the Greek King Menelaus (Prologue, 8-10). What? You say the cause of the Trojan War is more complicated than that? Well, not according to Troilus and Cressida. In the play, Shakespeare doesn't have a lot of patience for this guy—he portrays him as a selfish brat who cares more about getting laid than the people who are killed fighting in the Trojan War. As his dad points out, Paris acts "Like one besotted on [his] own sweet delights" (2.2.142).
Case in point. Paris doesn't mind sending soldiers to fight for his right to keep Helen but he doesn't spend nearly as much time on the battlefield as he should. At the beginning of the play, we hear that he's sustained a minor injury in battle (1.1.11-112), but, for the most part, Paris spends all his time making googly eyes at Helen (3.1.30-33) and getting jealous when she flirts with other men (1.2.166-167). With a guy like that for Prince, no wonder the Trojans lose.