Most puzzles are "fully interlocking" - that means that adjacent pieces are connecting such that if you move one piece horizontally you move all, preserving the connection.
A large number of fully interlocking puzzles have all the pieces of a similar shape, with rounded tabs out on opposite ends and corresponding gaps on other sides to receive the tabs of adjacent pieces. Usually jisaw puzzle pieces have four sides but there are puzzles with pices that are unregularly shape. The uniform-shaped fully interlocking puzzles are considered to the be the most difficult to solve, because of the subtle differences in shapes and sizes between pieces.
Some puzzles also have pieces with non-interlocking sides that are usually slightly curved in complex curves. These are considered to be the easiest to solve.
Most jigsaw puzzles are square, rectangular, or round, with edge pieces that have one side that is either straight or smoothly curved to create this shape, plus four corner pieces if the puzzle is square or rectangular. Some jigsaw puzzles have edge pieces that are cut just like all the rest of the interlocking pieces, with no smooth edge, to make them more challenging. Other puzzles are designed so the shape of the whole puzzle forms a figure, such as an animal. The edge pieces may vary more in these cases.
Since the earliest days of jigsaw puzzles the manufacturers have constantly endeavoured to create new cutting styles that differentiate their work. Even amongst modern, mass-produced puzzles there is considerable variation in the size, shape and intricacy of individual pieces.
The method of cutting pieces varies from puzzle line to puzzle line. Two puzzles of the same size and series from the same manufacturer usually have exactly the same cut, since the cutting dies are complex and expensive to make and so are used repeatedly from puzzle to puzzle. This enables disparate puzzles to be combined in odd ways. Larger puzzles are commonly cut into two or more sections.
More recently, technology such as computer controlled laser and water-jet cutting machines have been used to give a much wider range of interlocking designs in wood and other materials. These methods, however, have the undesirable effect of removing a small amount of material giving a loose fit with the adjoining pieces.
Beginning in the 1930s, jigsaw puzzles were cut using large hydraulic presses which now cost in the hundreds of thousands of dollars. The cuts gave a very snug fit, but the cost limited jigsaw puzzle manufacture only to large corporations. Recent roller press design achieve the same effect, at a lower cost.[Source]