External filters are popular because of how easy they are to install and maintain. These come in two different varieties, Pressurized and Unpressurized. Both types commonly incorporate a UV Clarifier which is one of the easiest and effective algae control methods.
The unpressurized filters are trickle-style filters where the water enters from the top and slowly trickles through the filter media and back into the pond. These are generally very easy to clean; open the top and there is a pad that you can pull out and rinse. They can be unsightly and hard to camaflouge. They generally sit right on the edge of the pond so they can take away from the beauty of the water garden as a whole.
The pressurized filters can be buried in the ground or placed a good distance from the pond because the output can be sent uphill. This makes them much easier to hide than the unpressurized versions, but they are more difficult to clean because the system has to be well sealed. It can be highly annoying to close the filter, then realize you have a drip leak because some debris got onto the o-ring. In most cases you can solve that by opening it back up, wiping the seal, then closing it up again.
These can easily incorporate a UV Clarifier. They also are much easier to clean than filters that sit on the bottom of the ponds. They are also relatively cheap and they are very easy to install.
External filters are harder to hide than other types. They also need to be removed in the winter to keep them from freezing.
External filters are ideal for small to medium ponds and they are very easy to install and maintain. They do take away from the overall look of the pond so they may not be desirable for some owners. Most are too small to be effective in large water gardens but they are easy to add on after installation so they are commonly added as a secondary filter.