The word "ship" (which is slang for "relationship") is a commonly used term online. Here are some popular Harry Potter ships.
This ship has basically been confirmed by JK Rowling, and is very beloved among fans. One of the first signs of a Ron/Hermione relationship was in their 4th year. Ron became insanely jealous to find out that Hermione was going out with Viktor Krum, the star seeker of the Bulgarian Quidditch team. He was constantly fighting with Hermione throughout the whole year, and took a strong disliking to Viktor, who had once been his all-time favorite Quidditch player. When Viktor leaves at the end of the year, things die down a bit. But in their 6th year, we begin to see even more signs. Ron finds out from Ginny that Hermione kissed Viktor in their 4th year, and begins to give a confused Hermione the silent treatment. They begin to fight more than ever. In order to get back at Hermione for dating Viktor, Ron starts to date Lavender Brown (even though he doesn't really like her). Hermione, in turn, begins to date Ron's Quidditch rival Cormac McLaggen to make him jealous. Eventually, Ron becomes too annoyed with Lavender and breaks up with her. Hermione soon does the same with Cormac. In the 7th book, Ron and Hermione become closer than ever after a rough patch. Eventually, the two end up married with two children, Hugo and Rose.
When Harry first met Ginny in his 2nd year, she had a huge crush on him and couldn't even talk to him or be in the same room with him. After her saved her life in the Chamber of Secrets, Ginny eased up a bit around Harry and they became just friends. Ginny eventually went on to date other people, including Michael Corner in her 4th year. But in Harry's 6th year, the tables begin to turn. It all starts when Harry and Ron walk in on Ginny and her new boyfriend (Dean Thomas) kissing in the corridor. Harry becomes really angry, but tries to convince himself that he is just feeling protective of Ginny, sort of like an older brother. Eventually he realizes that this is not the case and that he really likes Ginny. But he doesn't say anything about this to anyone, especially Ron, because he feels they'll disapprove. But one night after a Gryffindor Quidditch victory, Harry cannot contain himself and kisses Ginny in front of the whole common room. Ron appears to be okay with it, so they start to go out. Things are going great for the two until Dumbledore's death. It is then that Harry realizes that going out with Ginny is putting her in too much danger. After the break-up, both Ginny and Harry are devestated. In the 7th book, Ginny keeps the DA going strong at Hogwarts while Harry hunts for the horcruxes. After he survives the final battle, Harry gets back together with Ginny. Eventually they get married and have three children named James, Lily, and Albus.
This ship was a very popular one for a while until JK Rowling confirmed that it would never happen in an interview. Now we definitely know that Harry and Hermione are just really close friends, nothing more.
I bet that this ship is purely a work of fandom. When asked about it, JK Rowling has said that it is an "interesting" idea, but will not happen. Supporters of this ship say that Draco's constant teasing and talking down to Hermione are just signs of affection, but personally, I think he's definitely gone to far for it to be affection. Hermione is a very proud person, and she respects herself too much to go out with someone who insults her and her family.
This ship is another that is probably just for fan fiction. I have never heard JK Rowling questioned about this, but I'm sure it is as unlikely as Draco/Hermione. Supporters say that this is the classic Romeo & Juliet scenario. Ginny and Draco are from families who hate each other, and would therefore; have to keep their relationship private. I think that this is ridiculous, though. Ginny has no feelings for Malfoy except loathing, and he feels the same way about her.
This ship was very prominent in The Order of the Phoenix, but JK Rowling has made it known that it is officially over. Harry has liked Cho since his 3rd year, and he even asked her to the Yule Ball in his 4th year (although she had to turn him down because she'd already agreed to go with Cedric Diggory). In Harry's 5th year, after Cedric's death, Cho is extremely sad and distressed about losing her boyfriend, and she is struggling with her feelings for Harry. The two do end up going on a date, but it is brought to an abrupt halt when Harry mentions that he has to meet Hermione later that day. Cho becomes jealous of Hermione and storms out on Harry, who has no idea what he has done wrong. They later make up, only to be brought apart again by Cho's friend Marietta.
Viktor Krum, the Bulgarian Quidditch Player, went out with Hermione and escorted her to the Yule Ball in her 4th year. After he went back home, the two continued to correspond through the mail, much to Ron's dismay. In the 7th book, Hermione and Viktor meet again at Fleur's wedding, but it is apparent that they are just friends.