They have a history; a relationship. What kind of relationship depends on who you ask. He was the middle-aged gruff military perfectionist who seemed to lack a certain approachable quality. While she was a 30 something intelligent free spirited everything should be fun woman. In an honest comparison, they were complete opposites.
To most of the team it was somewhat of a mystery. Tony, Ziva, and McGee had discussed it many times trying to figure out exactly what it was that was between them. Sure, Gibbs was extremely protective of her, complemented her more than most, consistently provided her with her favorite beverage, soothingly rubbed her back and gently kissed her cheek after a job well done. He also scolded her for her ramblings, and pierced her with his trademark stare when needed, albeit her portions of these seemed to be less intense than what the rest of the team receives. Abby, well, she was the only one that he really opened up to and seemed to relax around if he ever really did either of them. His intimidations didn't seem have the same effect on her as they did the rest of the team. That could be explained in the viewing on his delivery method. He was never quite as stern with her as others. She seemed to read him, and know him in a way that most didn't and wouldn't want to, or weren't allowed to.
Regardless, of the different paths their discussions lead them down, they always ended the same. Tony and Ziva doubted any type of relationship that exceeded the boundaries of a close friendship for three reasons. One, they knew that both Abby and Gibbs have dated other people in the past; some serious, and some not so much. Two, because they didn't think it would be possible for them to conceal their feeling for each other around the rest of the team without them being discovered; after all they were surrounded by trained Federal Investigators all day. Lastly, and actually quite simple, taken directly out of the Gibbs rulebook: Do not date your co-workers. Regardless of their reasoning, Tony and Ziva though it would be a bit strange, but overall were ok with whatever it was between Gibbs and Abby, while McGee admittedly rejects the idea as absolutely preposterous. It's never been spoken, but it is clearly visible that McGee still has unresolved feelings for Abby, and never really moved on after she broke it off with him.
Ducky, well that was an entirely different story. He was not aware of their entire history, as anytime he broached the subject with them it was swiftly changed, but he was well aware of their current feelings. He was never positive that a sexual relationship had transpired, and he never talked about it with the rest of the team, but he knew that Gibbs was in love with Abby even if he didn't. He has known Gibbs for many years and has watched the women and marriages come and go, but what he has never seen before is Gibbs look at a woman like he looks at Abby. To Abby, Duck was a father type; he always listened, offered advice, and never judged. Although the direct conversation of "…are you two…" never came up, he could see her fondness of Gibbs as well.