"Red Dragon" isn't exactly a remake of "Manhunter", but they are both adapted from the same novel - kind of like "Dangerous Liaisons" and "Valmont".
The book "Red Dragon" was the first of Thomas Harris' works to feature Hannibal Lecter. It was published in 1981. "Manhunter" was made a few years later (1986), but didn't achieve that much notoriety. The book "The Silence Of The Lambs" came out in 1988 - written as a sort of sequel to "Red Dragon". The film of that (1991), of course, was immensely succesful. It was made by completely different people from the ones involved with "Manhunter" though.
Everybody wanted to do a sequel. That eventually came along in the form of the book and (shortly afterwards) film named "Hannibal" (book in 1999, film in 2001). In a totally shameless act of commercialism they then decided to make another film based on the original novel, "Red Dragon", but allowing Anthony Hopkins to play Lecter this time round.