Origin: Both claim the island as their own, and both have had some history together on the Island.
Prize: The island itself.
Fuel: Widmore lived on the Island as an Other and their rivalry began when Ben brings Alex into their camp. Widmore is mad because he told Ben to kill Alex but Ben decided not too. ("Dead Is Dead"). A war soon erupted triggered by Widmore's exile for fathering Penelope with an outsider, which is against the rules that the Others have. ("Dead Is Dead"). Ben is well aware that Widmore is searching for the island and that he is behind the freighter mission. ("Confirmed Dead"). Widmore claims that Ben has taken something from him that is not rightfully his. He is most likely refering to the island. Ben claims that Widmore changed the rules of their rivalry when Keamy executed Alex and declares his intention to kill Penny. Widmore and Ben then rise up the stakes on who can secure the Island first. ("The Shape of Things to Come").
Since Then: Daniel's mother never told him that Widmore was his father. Widmore followed his son's progress nonetheless, and he offered him a research grant when he graduated from Oxford. Years later, once Daniel's experiments had damaged his brain, Widmore offered to send him to the island to heal. Like Eloise, Widmore sent him to the island knowing he would die.
Origin: Desmond asked for permission from Widmore to marry his daughter, and he refused, even greatly demeaning him.
Prize: Penny's wishes
Fuel: Desmond asked for permission to marry Penny, but while Widmore admitted it was noble gesture and pulled out a very expensive bottle of MacCutcheon whiskey and says that unlike himself and the whiskey's maker, he is not a great man. Since Desmond is not worthy of drinking his whiskey, he is not worthy of marrying his daughter. ("Flashes Before Your Eyes") Desmond goes to Widmore at a auction and asks for Penny's number, which Widmore gladly gave to him since Desmond was a coward and his daughter now hated him. ("The Constant") After Desmond got out of prison, Widmore waited in his limo outside of the compound and offered Desmond a ride. Desmond reluctantly did. Widmore presented Desmond with 2 boxes, one containing all the letters he had written to Penny over the years, and the other contained an extremely large sum of money. Desmond refuses to take the money, despite what Widmore thought he would do since he was a coward. Desmond later participates in Widmore's race around the world in order to show him how honorable he is and win back Penny's heart, but Desmond instead gets lost at sea and crashes on the Island. ("Live Together, Die Alone, Part 1") Desmond later makes it off of the Island with the Oceanic 6, and Jack warns Desmond to not let Widmore find him. ("There's No Place Like Home, Part 2")
Resolved: Yes. Later Desmond finally married Penny and Widmore has urged them both to keep themselves hidden because of the war with Ben. ("Jughead") However Widmore kipnapped Desmond and brought him back to the island against his will but also told him he was sorry for doing so after he was attack by him for doing what he did. He then unleased a huge amount of electromagnetism on Desmond to see if he could survive, hoping Desmond would venture, as he successfully did, into The Heart of the Island and stop the Man in Black from destroying it and escaping. As proven in the flash-sideways timeline, now known to be the Afterlife, Desmond and Charles Widmore seemed to have a very good working relationship, symbolizing that each were able to accept their differences and "move on."
Origin: Both were members of the Others in the early 50s
Since Then: The two rose to levels of authority within the Others' hierarchy, eventually becoming leaders. They also became lovers and conceived a child together. Eloise left the island, perhaps to deliver the child, and Widmore had a second child with another woman. After being banished from the island, he then spend 20 years trying to return, not realizing that Eloise, whose whereabouts he knew, knew how to return all along. Both lived in London for a period, and both eventually conspired to send their son Daniel to the Island.
Origin: During his week aboard the Searcher, Jack learned Widmore sent the freighter that attacked the survivors.
Since Then: Jack had already decided that the Oceanic 6 had to lie to protect those who stayed behind. Once Jack learned of Widmore's involvement, this became a quest to avoid him, and Jack's last words to Desmond before they parted were a warning to flee from the man. Years later, Jack traveled to 1970 and met Widmore when he led the Others. Charles knocked Jack out with a rifle butt and held him captive till Jack convinced Eloise to follow Jack's plan.
Both ended up on the island in 2007, but though the periscope of Widmore's submarine caught sight of Jack on the beach, the two didn't cross paths.
Origin: Locke interrupted 17-year-old Charles from attacking somestrangers, presumably army personnel. Disarmed and vulnerable, Charles fled.
Since Then: Widmore was surprised to find Locke able to track him. He was further surprised when Locke knew his name, and a short time later, this mysterious self-proclaimed leader of the Others vanished into thin air. Widmore may have kept tabs on Locke over the years. Some decades later, DHARMA's soon-to-be security head claimed to be on the island to wait for Locke, and Richard left the island several times to recruit him. After Widmore was banished, Abaddon tried to draw Locke to the island, and by 2004, Abaddon worked for Widmore. Locke, meanwhile, learned of Widmore on the island from a video of Ben's before traveling back in time and seeing him in the 1950s. Widmore finally got to see Locke again in 2007, when Locke appeared an the Tunisia site he monitored. Charles recruited Locke to bring his fellow survivors back to the island. Locke died during this mission.
Origin: Windmore encountered the monster during his years as the Others' leader
Since Then: Widmore didn't understand what the monster was during his tenure. Years after he left the island though, Widmore said Jacob visited him and recruited him to fight the Man in Black. Widmore piloted a submarine to the island, set up a sonic fence to block the Man in Black and raided the Man's camp. And he brought a weapon against the Man in Black - Desmond. The two faced off on Hydra Island to no effect. They met again some days later in Ben's house, and the Man threatened Widmore into explaining Desmond's purpose.
Origin: Widmore fathered Penny off-island while still the Others' leader. This contributed to their banishing him.
Since Then: The two appeared to have a rough relationship for some time. In 1996, Penny described a potential life of working under her father as a "miserable existence". Widmore tried to protect his daughter, discouraging Desmond from marrying her. Some months later, he gave Desmond her address so she could express her feelings herself, and during Desmond's time in prison, Widmore refused to forward his letters to her to prevent the relationship from resuming. Widmore and Penny appear to have shared some resources over the next few years. Soon after Penny's listening station located the island, Widmore was able to dispatch a freighter there. By the time Penny rescued the Oceanic 6, however, she realized how dangerous her father was, and she went into hiding to avoid him. Widmore, meanwhile, learned Ben planned to kill her, and he eventually approved of her hiding and her relationship with Desmond - if it kept her safe. Widmore's last act before dying was to talk to the Man in Black to try to save Penny from him.
Though they were father and daughter and appeared in seven of the same episodes, Penny and Charles Widmore never shared a scene.
Origin: Sun did not know Widmore until she left the Island, though coincidentally, she once used a pregnancy test that was made by Widmore Labs
Since Then: Though Jin supposedly died on Widmore's ship, Sun blamed Ben for his "death", and she sought out Widmore's help because they shared a common enemy. She first approached him openly on the street, and this audacity shocked the man. Widmore later had her held at London's Heathrow airport, where the two conspired to kill Ben. Once Sun reached Los Angeles, Widmore posted her several surveillance photos of Ben - and a gun.
Sun didn't end up killing Ben when she found out Jin was alive, and she instead traveled with him to the Island in 2007. Widmore ended up coming to the Island as well via a submarine, and he retrieved Sun's camera from her crashed plane. He later imprisoned Sun along with Jin and several followers of the Man in Black in cages, therefore, he was ignoring their earlier off-island partnership. Widmore was later killed by Ben at the Barracks while Sun died with her husband Jin on Widmore's submarine.