So you are tired of the manual cleaning process of your swimming pool. Thinking about buying a new pool cleaner, or replacing a worn out one? Here is a basic primer on the pool cleaners.
Suction Side - Usually the cheaper priced models because they have less moving parts. These either hook into your skimmer via an attachment, or a dedicated suction side port on your pool. The dedicated port is a much better way to go on this as it does not render your skimmer useless while the cleaner is in use. Since the water is being sucked in (think vacuum cleaner) the debris is carried with the water to your filter where it is removed.
Types with Wheels or Feet (Examples: Hayward navigator (feet), ThePoolCleaner (wheels)) suck water and moves by a turbine inside the cleaner, causing it to spin, which drives the cleaner. Use a programmed drive box to randomly cover the pool.
Disk Type (Examples Kreepy Krawly, Barracuda) use a valve to open and close the water flow, causing the unit to jump (looks like vibration). Unit follows the hose around the pool.
Pressure Side - Cheaper models use your pools return (Polaris 360 as an example) to power them. Better models use a dedicated pump. As in the suction side cleaners, the moving water spins a turbine (depends on brand, but that is basically it) or uses water jets for movement. These usually have an external bag, that jets of water cause the debris to wash up into. The amount of debris the bag can "catch" is limited by the size of the weave, usually allowing smaller particles to pass through. Some have a tail whip used to stir up this small debris so your pump will draw it in via the inlets and the filter will catch it. Use a programmed drive box to randomly cover the pool.
Robotic - These are typically self contained cleaners with their own motors and pumps. They have brushes (basically) that spin and use water jets and bags like the pressure side cleaners. These plug into an outlet by the pool, and reduce the voltage to a safe level, and run off electricity. Some battery operated models exist. Use a programmed drive box to randomly cover the pool.
In Floor - Little jets pop up on the pool floor, and force all the debris towards the main drain where it is then forced through the filter. Depends on installation placement to cover the whole pool.

Pros - Cons (I will cover the biggest ones I know of):
Suction Side:
 Pros - Usually cheaper, nothing else to buy, do a good job on fine debris like sand and dirt. Can leave in the pool all the time except when swimming.
Cons - Not so good for leaves, twigs and acorns. Can make your skimmer useless during use without a dedicated suction port. Uses main pump, can only run when it is.
Pressure Side:
Pros - Cleans leaves, twigs and acorns. Can run without the main pump running. Can leave in all the time except when swimming.
Cons - uses additional electricity if you use a additional pump. Does not get sand and dirt as well.
Robotic:
Pros - With the proper bag, can do the best job on all debris in the pool. Scrubs the walls (the brushes thing).
Cons - COST, have to place in pool, let it clean, then remove. Weight during insertion and removal.
Floor:
Pros - don't have do anything, they come on with the main pump, go off when it shuts down.
Cons - Have to be installed in the floor, usually during building. Main Drain does not get leaves and twigs, have to manually take those out.

I have found over the years that people become passionate about their pool cleaner of choice, like cars or sports teams (not sure why, just one of those things I guess). I personally have used pressure and suction side cleaners, and am partial to the suction side with a dedicated suction port. But I like to leave the cleaner in all the time (except for swimming) so those two are a better option. A couple people who have the in floor cleaners like them, but that is not something a current pool owner can buy. People with robotics really seem to like them, but they are more work, and I can buy 2-3 suction side cleaners for the price of one robotic. Plus since my pool is outdoors, it is always getting something in it. Leaving the cleaner in all the time means I always have a clean pool. There you have it, that should at least give you a little background on what you might in for. There are newer models coming out all the time, some swim/float instead of roll, vibrate or walk. If you are having a pool built, talk to your builder, see what they recommend for your environment. Just make sure you are getting the proper cleaner. Above ground pool cleaners are smaller and cheaper than inground pool cleaners for a reason. If it is not for a lined pool, and you have a liner, be careful, make sure you get one for a lined pool.
Good Luck