Priam is the King of Troy and the father of Paris, Troilus, Hector, and Helenus. In the play, he's portrayed as a loving but overindulgent dad who lets his sons have the final say in everything. It turns out that being a wimpy dad / king has some pretty tragic consequences. Like total ruin.
Case in point, he tries to talk some sense into the princes about the folly of fighting a deadly and costly war just so the Trojans can keep Helen. At the council meeting, he says, "Paris, you speak / Like one besotted on your sweet delights" (2.2.143-144). Translation: Paris is acting like nothing more than a selfish, horny kid who's willing to risk the lives of his soldiers just so he can keep getting it on with Helen. But, by the end of the meeting, Priam basically backs down and just goes along with the war, which, by the way, Troy totally loses.
Here's another example of Priam's unwillingness to take a firm hand with his sons. After hearing the prophecy of Hector's death, he begs his son to stay away from the battlefield but then relents when Hector blows him off: "Farewell: the gods with safety stand about thee!" (5.3.94). As we know, Hector is killed in battle, and the play predicts that the news of Hector's death will be so devastating that it "will Priam turn to stone" (5.10.18).
Gee, maybe you should have said "no" once in a while, dad.