They can move large volumes of air at very low pressures. The other type is a centrifugal unit, and these are similar to a centrifugal water pump. They have a plate with radial vanes and the air is forced out by the centrifugal force. The latter is rarely used in domestic fans. They are often used on furnace blowers, and similar applications where bulk is not such a problem, and you need high pressures.
Axial fans are the most common, as they are the cheapest to produce and more compact.
One disadvantage they have is that they cannot develop high pressures, with a limit around 60Pa-80pa for household sizes. Furthermore, if they are run with to little air, such as by using ducting that is undersized, then they become noisy as the air cavitates passing the blades. Centrifugal fans with the same size inlets as axial fans can develop pressures up to 4 times that of an equivalent axial fan, however, they also usually cost a minimum of twice that of an equivalent axial fan. Like all things, quality costs money. If you are using a window or wall fan with integral shutters, get one with silent opening.