The couple in the Harry Potter series that I ship is Molly and Arthur. In my opinion, Molly and Arthur are the most overlooked and under appreciated couple in the books. There's very little fan fiction about the two of them, and at least half of that fiction ships them with someone else, not each other. However, that's not why I ship this lovely couple.
Arthur and Molly were made for each other, and its very evident. They have seven children that could have only been produced out of their love for one another, and even when times are hard, we don't see them being hostile toward each other, instead, they turn to one another for comfort. They may not be the wealthiest of the wizarding community, but they get by, and what they don't have in money, they make up for in their love. Arthur is the love of Molly's life, and in essence, they are the perfect couple. Arthur is also the greatest source of Molly's frustration, as he does do things behind her back that he knows will make her angry with him. He doesn't ever mean any harm by the things he does, and Molly knows that, but even so, I'm fairly certain that Arthur does a fair share of groveling and begging to gain her forgiveness.
Arthur does his best to provide for and support their brood, and he doesn't mind living in poverty when the alternative is that he is in a job he hates, and possibly Molly working as well, instead of staying at home to raise their children in a loving environment. I think that Molly chose to work in the home, she seems extremely happy to be raising her children, and she takes pride in the fact that she does run the household and has dinner waiting for her husband when he gets home after a long hard day.
As a couple, Molly and Arthur have been through quite a bit together, having lived through the first rise of Voldemort and now fighting in a second war against him. When Molly lost her brothers in the first war, I believe that Arthur was there, and he did his best to make sure that she didn't dwell on her grief and to show her that even though she had lost two people who were no doubt important to her, she still had him and their children. I also believe they are a stronger couple because of all that they have been through, and stand taller because of all the trials they have survived as a married couple. I firmly believe that Molly would be absolutely devastated if she lost Arthur, and I can only imagine what she felt the night Dumbledore wrote to her and told her that Arthur had been attacked while on guard duty for the Order, and how she felt when they were separated for a length of time so he could heal in Saint Mungo's. Also, I'm certain that Arthur would be a right wreck if something happened to Molly, as they are roughly the same age, and he has probably been in love with her since they were sixteen or seventeen years old. He also probably would have no idea how to take care of himself, and he would be broke within a matter of months as we saw in Chamber of Secrets that he wanted to take the Grangers to the Leaky Cauldron for a drink when he knew that he and Molly didn't have much extra money.
So, even though there are plenty of adult couples in the series that do get the glory such as Remus and Tonks, and other pairings, Molly and Arthur are by far the model couple, and its sad that they're mostly overlooked.