He is an antihero in the first film, an antihero turned minor character in the second film and an important character in the third film.
The Curse of the Black Pearl
He is the governor of Port Royal and the father of Elizabeth Swann. In contrast to his strong-willed daughter, he is something of a milquetoast. Both Elizabeth and the Governor sailed from England to the Caribbean eight years prior to the events of Curse of the Black Pearl, along with then Lieutenant James Norrington, presumably when Swann assumed the governor's post. Swann is a doting father, and he wishes for his daughter to accept Commodore Norrington's marriage proposal. However, he eventually comes to accept that she truly loves Will Turner. According to the Pirates of the Caribbean Visual Guide, Elizabeth's mother died when Elizabeth was a young child. It is possible that Governor Swann holds some sort of peerage, as his butler refers to him as M'Lord. Many Colonial Governors held peerages or knighthoods, however, it is not directly stated in the films whether Swann holds one.
Dead Man's Chest
In Dead Man's Chest, he protested a the fact that Will and Elizabeth were arrested by Lord Cutler Beckett for freeing Sparrow and Swann informed Beckett that Former Commodore Norrington had resigned from the Navy, after Beckett freed Will so he could go search for Jack Sparrow, Elizabeth escapes jail with help from her father Weatherby Swann but he is captured by Ian Mercer while Elizabeth is sent off by Beckett to offer Letters of Marque to Jack in return for the compass after threatening Beckett with his life. Later on Port Royal Beckett informs Governor Swann about his Daughter's status and that a EITC ship are in chase of the Black Pearl and that Elizabeth if caught will be saved. Beckett also informs Weatherby that he will be freed as long as he gives good reports to England about Beckett's missions.
At world's End
Weatherby Swann is forced to use his authority to allow Beckett to execute hundreds of pirates. In a deleted scene, Swann comes on board the Flying Dutchman when Beckett confronts Davy Jones about the attack on pirates fleets, Swann asks Jones if he offered the pirates a chance to surrender as Swann was concerned if his daughter was on board one of them, Jones informed him that his daughter is dead as she apparently went under with the Black Pearl which it was pulled down to the bottom of the sea by the Kraken. Horrified and driven temporarily mad with grief, Weatherby attempted to stab the heart of Davy Jones with a knife but Admiral Norrington stopped him and Jones informed him about the consequences of stabbing his heart. Beckett and Ian Mercer informed Swann that his daughter was still alive, and finally after Norrington plead for him to, Swann dropped the knife and walked away. After this however, due to him having a fairly good knowledge of the Dead Man's Chest and his attempt at stabbing it, Beckett ordered Swann to be killed and informed others such as Admiral Norrington that Swann was going back to England. He later appeared in Davy Jones' locker informing the crew of the Black Pearl about the chest and ignored Elizabeth's request to come aboard the Pearl, as he had truly died. His final scene is when he promises to give Elizabeth's love to her mother, and sails away into the locker. Though Elizabeth attempts to rescue him, Will Turner, her fiancé, stops her and Tia Dalma confirms that there is no way to save Governor Swann, as he is at peace.