Traditionally, the 'correct' sitting style in Japan is seiza, kneeling with the weight on top of the lower legs, which are folded underneath the body. However this can become painful after long periods of time or for people who are not used to it, so many prefer the more comfortable zaisu, where the back is supported and legs can be sprawled in front.
Zaisu come in many styles, and can either have a cushion built in or be used with a zabuton. For an even more relaxed seating arrangement, one may also use a Japanese style armrest called a kyōsoku (脇息?).
Zaisu are very common in Japan, particularly in houses with traditional Japanese-style rooms (washitsu) where low tables and sitting on a floor of tatami mats are commonplace. They are perfect for relaxing on a cold night under a heated kotatsu table.