Captain Swan is the ship between Emma Swan and Hook from the Once Upon A Time fandom.
After their kiss in the last part of the episode 'There's No Place Like Home' Captain Swan (Killian and Emma) officially became canon after Emma kissed him outside Granny's diner after Hook told her that he traded his ship for her in order to save her family.
In season 4 they start dating and hit huge milestones in their relationship such as Killian revealing to Emma she is his happy ending or the large, more heart wrenching one when Emma finally admits to Killian that she loves him before getting sucked into the vortex of darkness.
Their relationship progresses in season 5 where not only do they exchange the words 'I love you' many times, but Killian also gives Emma Liam's ring showing his devotion towards her. Another adorable Captain Swan moment is when we find out that Killian bought Emma a house that he hoped they would share with Henry. However they don't get to have the 'white picket fence life' they dreamt of as Killian makes the ultimate sacrifice for Emma and makes her kill him in order to save her and her family. Emma wont let her true love stay dead for long though and travels to the underworld to follow in the footsteps of her parents and split her heart in half, proving their true love and bringing Killian back to Storybrooke.
They first meet when Hook is under cover working for Cora. Emma is Hook's enemy.
Hook guides Emma and her team through Neverland and he obviously has a crush on her even telling her so. However, he is constantly rejected but at one point they did kiss (Good Form 3x05) but it was only gratitude although it was extremely passionate. They become friends after they return successfully from neverland. When Emma loses her memories hook tries to see if kissing her will trigger anything but it doesn't. After she gets her memories back they become better friends. Zelena curses hook's lips and tells him to kiss Emma as it will strip her of her powers. Zelena harms hook and tells Emma that the only way to get him back is to kiss him. She does so and it works but she is now powerless. When they are sent back in time they have several intimate scenes including when she has to make out with his old self in order to keep him occupied. After they get back and Emma learns that he sacrificed his ship for her she kisses him of her own will.
In the beginning of this season Emma pulls away from Hook. Hook confronts her outside of Granny's in episode three and they have a very sweet moment ending with a passionate kiss. Hook and Emma are now in a relationship and in the season finale Emma reveals that she loves hook to Regina after hook is 'killed' in an alternate universe, which was never really real. Emma later then is unable to tell hook she loves him but at the end when she sacrifices herself for Regina she finally tells him before becoming the dark one.
When Emma becomes the dark one, she is transported to Camelot, so everyone travels there to save her. They meet up with Arthur, Guinevere and Merlin. Merlin says he can take away Emma's darkness. He fails, and Hook gets hurt badly and Emma turns him into a dark one, also, to save him. She creates a curse to bring everyone back to Storybrooke and she takes away everyone's memories. Soon, though, Zelena gives Hook back his memories and he is extremely angry at Emma. Emma does all she can to stop him from doing bad things but fails. Hook summons all the past dark ones from the Underworld. But at one point, as Nimue, a past dark one, is harming Emma, Hook comes to his senses and sucks all the dark ones back into Excalibur, but he tells Emma she has to kill him with it so all the darkness will be destroyed forever. It was the saddest Captain Swan scene ever. They share a kiss and then Hook dies. Emma goes to see Rumpelstiltskin later and finds out the darkness was not destroyed but transferred back into Rumple. A pissed Emma decides to go to the Underworld to bring him back.
Emma and Hook's love is confirmed to be true when the two try to use the True Love Scale. Emma chooses to save Hook instead of her own heart, making them pass the test.
At the end of the season finale (5x23 "An untold story") Emma decides to say those three little words to Killian with out any battles or death between them, that " I love you" that Killian and all the captain swan fandom had been waiting for.
Since Hook was introduced many 'Oncers' have been shipping Captain Swan due to their witty banter and common innuendos. In season 3 it became apparent that Hook has a crush on Emma causing the CaptainSwan fandom to grow even more due to their sentimental moments and more importantly their first kiss in Neverland even though Emma insisted it was a one time thing. However, the ship had a rivalry with SwanFire, many debated whenever Emma should be with Neal as he was Henrys father though that soon was no longer an option when Neal died. Soon Captain Swan gained even more followers and became the most loved ship in the entire fandom.