Short answers, pulled from the story.
Denethor II orders his servants to burn him alive on a funeral pyre prepared for himself and Faramir. He breaks the white rod of his office over his knee, casting the pieces into the flames before dying while clasping the palantír in his hands.
Sauron biases what Denethor sees through the stone, showing him only despairing visions such as the Black Fleet approaching while concealing that Aragorn's troops are coming to rescue them. This deception drives Denethor to suicide during the Battle of the Pelennor Fields when forces of Mordor arrive at the gates.
Denethor II serves as the Steward of Gondor who rules from the stone hall within Minas Tirith. Gandalf describes him as proud and subtle with far greater lineage and power than Théoden of Rohan, though he remains too strong to be corrupted directly by Sauron until the end.
Scholars compare Denethor's madness and despair to Shakespeare's King Lear regarding their shared grief over perceived loss of children. Both men are first outraged when their children refuse to aid them, then grieve upon their children's death which is only perceived in Faramir's case.
Peter Jackson chose to depict Denethor played by John Noble as greedy and self-indulgent quite unlike Tolkien's powerful leader. Tom Shippey comments where Tolkien's Denethor is a cold ruler doing his best for country Jackson's version looks greedy and gobbles a meal while son Faramir has been sent out in hopeless fight.