Rear Window was shot on 35mm film in Technicolor on a huge soundstage at Paramount Studios in Hollywood. Plenty of movies in the '50s were still shot in black and white. Technicolor was a very expensive process, first because the cameras were much more complicated than black-and-white ones but also because you needed the tech boys back at the Technicolor labs to properly develop the film for you. Rear Window was a big deal, with the most famous director in the world and two of Hollywood's biggest stars in its corner. No way they were going to make this in black and white, especially with Grace Kelly's gorgeous wardrobe.
Colors seem more vibrant with Technicolor. They pop a little more, and there seems to be more of a contrast between the bright colors (like red and yellow) and duller ones (like gray and brown). (Source) Even the nighttime scenes have a certain lushness to them, which adds to the slight sense of a dream (or an altered state of consciousness).
Hitchcock built the enormous set on the Paramount soundstage. The apartment/courtyard set was 98 feet wide, 185 feet long, and 40 feet high. There were 31 separate apartments, 12 of which were completely furnished, including electricity and running water. The actress who played Miss Torso even lived in her apartment during the entire filming. Jeff's apartment was actually at stage level. The soundstage floor was excavated, and the courtyard was actually about 25 feet below stage level. It took about 1,000 arc lights to simulate the sunlight for the daytime scenes. The lights got so hot at one point that they set off the sprinkler system. (Source)
The director stayed in the set of Jeff's apartment during the entire shoot and spoke to the other actors remotely through flesh-colored earpieces. Pretty, pretty, pretty clever.