The filter is a hang on the back (HOB) style of power filter, and possibly a Top Fin brand.
These filters work by sucking water into the filter case, letting the water flow through the filter media (the blue and black plastic thing with the black sponge), and the clean water flows back into the tank. The beneficial filter bacteria live on the black sponge. There might be a small mesh bag container ceramic or plastic beads that also holds beneficial filter bacteria.
The black sponge in the blue and black plastic thing might container carbon (black granules). Carbon is used to absorb chemicals from water and after a period of time it becomes full and should be removed and replaced if you want to replace it. However, most aquariums do not need carbon and it will absorb medications and plant fertilisers if you have it in the tank. I do not recommend using carbon unless you have chemicals in the water that need to be removed.
You can usually make a small cut in the top of the black sponge and shake the carbon out and throw it away. If you want to add new carbon then buy a box or bag of Activated Carbon or Highly Activated Carbon from a petshop, rinse some off under the tap to remove any dust. Then add it to the black sponge via the small hole you make in the top to remove the old stuff.
To clean these types of filter you turn them off at the power point and unplug them from the power socket. Take the filter materials out of the filter and squeeze them out in a bucket of tank water. When they are reasonably clean, get another bucket of tank water and squeeze them out in that. Put the filter materials in the tank. Pick the filter up and carry it outside and pour the old water onto the lawn or garden. Remove the intake pipe and then rinse the filter and intake pipe under some tap water.
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On most HOB style filters, you can separate the motor from the case by holding the motor (under the bottom of the filter case), and gently turning it 90-180 degrees and then carefully pulling the pump away from the case. Rinse the case where the motor joins it to remove any gunk, and wipe the motor down where it fits in the case.
In the motor is an impellor. It will be a thin magnetic cylinder about 2 inches long x a bit less than 1/2 inch in diameter, and have a plastic bit on one end with 3 or 4 plastic blades that stick out evenly around the magnetic bit. There might also be a small plastic fitting that sits on top of the impellor and that should have a rubber bit that goes on the end and a thin steel rod (impellor shaft) that the impellor sits on.
Remove the plastic fitting (if it has one) and the rubber end bit and grab the plastic blades and carefully pull the magnetic impellor out. It should be a little tight but will come out. The magnetic impellor will be pulled into the motor by the coil inside the motor. You can't see the coil because it is sealed up in the plastic part of the motor, but it is there and will try to pull the impellor back in. The impellor shaft might also come out (depending on the particular brand), some do some don't.
Once you have removed the impellor (and impellor shaft if it comes out), check it for small plastic washers and any rubber bits on the end of the impellor. Put these bits on a table where they won't get lost. Rinse the impellor, shaft and the inside of the motor under tap water. Then put the impellor, washers (if there are any) and rubber bits back into the motor.
*NB* The electrical components in the motors on aquarium power filters are normally sealed in a plastic resin and that is covered in black plastic, so they are usually safe if they get water on the outside.
Smear a thin layer of Vaseline on the o-ring (rubber seal) on the top of the motor and put the motor back in the filter case, turn and twist to lock it into place.
Put the filter case back on the tank and put the filter materials in the filter. Fill the filter with water from the aquarium and plug the filter into a power socket and turn it on, (make sure the plug is dry before plugging it into the power socket). Make sure the water level in the aquarium is higher than the motor on the filter. The filter will spend a minute or so sucking water into it before it starts flowing back into the tank. Check the filter for leaks around the motor.
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You can add a round sponge (from an internal power filter) to the intake tube on the HOB filter for additional filter media. You can also add sponges from other brands of filters to the HOB filter. If you buy a sponge you can easily cut it to fit your filter with a pr of scissors.
The following link shows an Aquaclear brand HOB filter being disassembled. Virtually all HOB filters are cleaned in the same way.
The second link shows you how to modify a HOB filter to increase the filter media in it.