Jigsaw puzzles are a staple educational toy.
Typical jigsaw puzzles are pictures printed on thick cardboard and cut into pieces in a slightly irregular pattern, so that they fit back together only the right way. Sometimes the cuts in cardboard are slanted so the complete picture is strong enough to pull it on the floor. Children usually can make pictures from two to four jigsaw puzzle pieces before their second birthday, up to twenty pieces around three years, hundred around five. There are much more complex jigsaw puzzles in hundreds and thousands pieces. I'm not quite sure who are supposed to make those. There is a great challenge for a toddler to use different clues to fit picture together, and still a big interest for a preschooler to take up a lengthy project and to recreate an interesting picture. The educational or creative value of re-making somebody's elses' mosaic in such a simple and non-durable medium escapes me.
Jigsaw puzzles interest children as far as children like pictures themselves. Some ABCDary jigsaw puzzles my kids received from relative strangers as birthday presents were never completed, while a space shuttle and a map of the USA were great success. Kids also seem to like big pictures, something taking over a living room.
See also: children, baby toys, children's toys, Duplo.