This answer is spoiler-iffic. Spoilers from books/movies 1 through 7.
So, at first, Ginny has a huge crush on Harry, the Boy Who Lived. She can barely talk to him. She always got embarrased and speechless when he was around, and even sent him an anonymous love poem in Book 2.
Sometime between Book 4 and Book 5, Hermione gave Ginny good advice to be more "herself" and start dating other people, and stop being so awkward around him. In Book 5, Ginny was much less shy.
During the summer before Harry's sixth year, Harry, Ron, Hermione, and Ginny were very close and at one point Harry started developing feelings towards Ginny. In a love potion, which smelt of what you were attracted to, he smelt "something flowery he might have smelt at the Burrow". His feelings are realized when he finds Ginny kissing Dean Thomas after a Quidditch match. For the next several chapters he has large inner debates, including whether his feelings are elder-brotherly or what Ron will think. He even at one point notes he is glad Ron could not perform Legilimency, a spell involving the reading of one's mind.
When Harry uses Felix Felicis, a potion that gives you lots of luck, that he won earlier in the year, he accidentally brushes Ginny in his Invisibility Cloak which Ginny mistook as Dean helping her in the portrait hole, causing a feud between Dean and Ginny. This wasn't his intent, but it lead to a breakup with Ginny.
Finally, after a Quidditch match, they kissed, Harry instigating it. Harry has some of the "happiest hours of his life" with her when they date, and the relationship is subject to much gossip. After the death of Professor Dumbledore, Harry breaks up with Ginny to protect her from danger.
The next summer, as Harry prepares to go on his dangerous quest to defeat Voldemort, Ginny kisses him passionately to remember her by. They are interrupted by Ron and nothing else happens. Harry is deeply in love with her, worries about her and, at the moment he thought that he was about to die, thought only of Ginny's blazing look and her lips on his. After the quest is over and Voldemort is defeated, Harry reconciles with Ginny. As it is shown nineteen years later, they get married, and have three children: James Sirius Potter, Albus Severus Potter and Lily Luna Potter.