A double stack magazine, also sometimes known as a 'staggered' magazine, is a box magazine for a firearm where the rounds are kept in two distinct columns (viewed from the front or back) in order to maximize the use of space. Since cartridges are (almost always) cylindrical, and magazines are oblong, a modest increase in width will allow the the rounds to be offset from side to side and allow significantly more ammunition to be carried in each magazine for the same vertical length of the magazine itself. This diagram won't show that due to the monospace font nature of it, but hopefully you'll get the idea:
Smaller handguns typically do not use this system, as the increase in width of the grip is undesirable, and the concealability or small size of the gun is more important than having more rounds available.
These magazines come in two important sub types - the single feed and double feed versions. Single feed magazines have an opening at the top which is the diameter of a single round, usually in the center of the magazine. This makes the feed mechanism inside the gun simpler, as the designer will always know where the round will be presented. However, the fact that the rounds must 'shift' as they rise in the magazine from side-by-side to single-flie means that not only are the magazines more complex (they taper), and carry fewer rounds, but that complexity of motion means that jams and magazine-related malfunctions are more common.
Double feed magazines will present the rounds at the top on alternating sides, as they feed. This means the feed mechanism in the gun must take this into account, but nearly all modern firearms were designed with this in mind and it's not a problem. Double-feed magazines are generally more reliable, especially for high-speed operations in automatic weapons.