
Kimagure Orange Road is the story of Kasuga Kyosuke, a kid who just happens to come from a family with psychic powers. When he comes to a new junior high school, he gets caught up in a love triangle with beautiful but moody Madoka--who seems to like him, but never lets on to anybody--and bubbly Hikaru--who definitely likes him, and makes sure everybody knows. For Kyosuke's part, he's terminally indecisive, and can't quite make up his mind between the two.
A Junior High School student at the beginning of the series, Kyosuke isn't particularly bright, and although he's a relatively nice guy, he's also about the most indecisive youth on earth, and has about as much spine as jell-o. He also has psychic powers; he can teleport, use telekinesis, and occasionally hypnotize himself. He also wishes he could read minds, but no luck.

A beautiful girl about Kyosuke's age, Madoka was hooked up with the wrong crowd for a while, and now has a reputation as being a bad girl. Known as "Madoka the Pick," most people avoid her, but she is actually much nicer than her generally reserved demeanor lets on. She sees something in Kyosuke, but has never let on to anyone. She has few friends, but is best friends with Hikaru.
Cute as can be and perpetually perky even though she's also a bit of a "bad girl," she is a little younger than Kyosuke and completely enamored with him. She'll do anything and everything to show her love for her "Darling." Although she is Madoka's best friend, she seems oblivious to her real competition for his heart.
Kyosuke's younger sisters; they also have psychic powers, and aren't nearly as careful about how they use them. Constantly terrorizing the family cat, Jingoro, and a perpetual nuisance to Kyosuke, they're also friends with Hikaru, and Madoka indirectly.
Kyosuke's loser friends. They will chase any cute girl who passes by, and are particularly interested in Kyosuke's sisters. And Hikaru. And...
A longtime friend of Hikaru and a martial arts expert with near-superhuman strength. He's enamored with Hikaru and can't understand why she prefers Kyosuke to him. He'll jump at any opportunity to defend Hikaru's honor, and would sure like the chance to beat up Kyosuke, too. Only present in the TV series, Yuusaku doesn't appear in the OAVs or either movie.
Kyosuke's family got its psychic powers from the maternal side, but his mother passed away a while ago. His father is a rather meek professional photographer who just sort of accepts his children's paranormal powers. Kyosuke's slightly lecherous maternal grandfather is around occasionally, and has all kinds of fun with his psychic powers. Various cousins, each with their own interesting psychic powers, also show up periodically.
The TV series is based on a comic book series by Matsumoto Izumi that ran between 1984 and 1988. The comics, however, follow their own continuity--there's no real connection to the animated version.
The times in the series match up roughly with when the episodes/OAVs/movies were created. The series starts at the beginning of the TV series, in about 1987, when the characters are in Junior High School. The OAVs more or less follow the TV series, and take place during Senior High School. The original series was concluded with the movie "I Want to Return to That Day" that takes place right before the characters enter college (though it was actually produced immediately after the TV series ended and before the OAVs came out), and there is a follow up movie, New KOR, produced in 1996 that takes place jumping between 1991 and 1994, while they are in college. The continuity in the series is not quite perfect--the OAVs are a little disjointed from the rest of it and technically the dates in the movie and TV series don't quite match up--but everything basically fits.
...But, in America, the order they were released was way off: AnimEigo released the OAVs and then the first movie quite some time ago, and after much fan encouragement also translated the TV series. Around the same time, AD Vision got ahold of the second movie, and also dubbed it (AnimEigo only ever did subs); the translation was close, but there are slight differences in the translated dialogue style.