Quantum Leap, on NBC
In 1999 (the series is set ten years after it began airing), Dr. Sam Beckett (Scott Bakula) was the head of Project Quantum Leap, a time travel experiment wherein he would step into the Quantum Leap accelerator and trade places with someone from some other time (within Sam's own lifetime). His body would travel into the past, surrounded by that person's aura, so that everyone would see and hear that person, not Sam. Meanwhile the person he displaced would appear in the future, surrounded by Sam's aura. These people would remain in the waiting room. There was a whole team of scientists and such to look after them. Also Admiral Al Calavicci (Dean Stockwell), who would go into the imaging chamber while the supercomputer, Ziggy, would get a lock on Sam's brain pattern, find out when and where he was, and recreate his environment in the chamber so that Sam and his surroundings would appear to Al as holograms. Meanwhile, a hologram of Al was sent back to appear to Sam and talk to him, give him information and help. Only Sam could see or hear Al, except occasionally for animals, small children, crazy people, and maybe someone with a brain pattern very similar to Sam's.
Anyway, Sam would have some mission he'd have to try to complete. See, the folks back home in the future couldn't retrieve him like they'd planned. So he'd just have to fix something about the past. Ziggy would do historical research to learn about what was going on with the folks Sam found himself with at the time he was there. He'd have to try and change history for the better. When he was done, he'd leap to some other person, place, and time. Quite at random. And Al and everyone back home would have to look for him again. Sam kept hoping to return home one of these leaps.
Well, there's your plot. It was a very fun show, and amusing, and clever, fairly original, sometimes quite moving. Sam and Al were great characters, very different people but great friends nevertheless. They worked well together. Sam was a brilliant scientist who knew pretty much everything, except that his brain was kinda swiss-cheesed, as Al said (memory lapses due to the leaps, but this mainly happened very early on), and he was very idealistic and ethical. Al, of course, was a Navy man, but not particularly formal. Very casual sort of guy, easily distracted, especially by women. I should probably say more about the series, but I'm afraid that's all I can think of at the moment....