Cartridges are a quick and convenient way of refilling a fountain pen. Most fountain pens these days take cartridges (with a few exceptions, such as Pelikan's Souveran range, TWSBI fountain pens, and the Lamy 2000). They generally work out a little more expensive than buying bottled ink, and fewer choices are available, but the ease of refilling beats those problems for most people. And if you are not one of those people? Most pens that use cartridges can also use bottled ink with the addition of a converter, giving you many of the advantages of both systems.
There are quite a few different types of cartridges, with the most common type being standard 'international' cartridges, which come in two different sizes. These are used by quite a few different pen brands, including most of the smaller brands. Quite a few of the bigger brands, though, have their own shape and size for cartridges, so you'll need the manufacturer's own cartridges if your pen was made by Parker, Sheaffer (with some exceptions), Lamy, Cross, Platignum, Platinum or Pilot (with some exceptions).