Advice on changing a washing machine pump
Purpose of the pump
The pump is just a small induction motor whose purpose is to empty the machine of water between the wash and rinse sequences and also during the drying cycle in washer dryers.
Faults and fault finding
The main symptom of a pump fault is that the machine won't empty. However, there are other reasons that can cause this so it's not wise to assume that the pump is at fault. If a machine won't empty it could be:
It is important to rule out blockages before deciding the pump is faulty as changing the pump will not cure the problem. Some machines have filters on the front, some have accessible filters but which are hidden behind a kickstrip (remove it and the filter handle can be found), and some have no customer accessible filter but there's often a filter in the sump.
Be aware that undoing a filter when the machine is full of water will result in the water draining out (!)
The main pump faults
The main reasons for changing a pump are:

Pumps these days come supplied as complete units so if there is a fault with the pump the only option is to change it.
Replacing
Pump changes are one of the easiest jobs on a washing machine. Most pumps are attached to the base of the machine by 2 screws or bolts and they have one hose connection to the sump and one to the drain hose. For most machines the best way to change the pump is to lean the machine almost on its side - it's best to rest it on something so it's not entirely flat. e.g. if looking from the front the pump is at the back right of the machine place it on its left side. Access is then relatively easy. Remove the old pump and fit the new.
Complications
When a pump fails the machine is usually stopped and full of water. The problem is getting the water out so that the machine can be worked on. If the pump has seized or stopped working there should be no blockages. In this case, placing the drain hose down into a bucket will allow the water to drain from the machine. If you attempt this and the water won't drain out then there's a blockage (a good diagnostic test for a pump failure vs blockage).
If the blockage is in the filter, one way to get the water out is to remove the filter as the water will drain out as the filter is removed. Quite a lot of water can come out so a suitable container needs to be used to catch it. One way of reducing the amount of water released is to tip the machine back 30 degrees or so.
Occasionally you will have an appliance that can't be drained and doesn't have a filter. In this case, all that can be done is to undo one of the internal hoses (such as the sump hose) and let the water gush out. If you can get the machine's door open you can scoop or syphon out some water from the drum, but this method inevitably results in a lot of hard to catch water being released. Disconnect the machine and do it outside if possible.
Difficulty
Easy for most machines. Some Zanussi built models have the pump fitted into the filter/pressure chamber but the method of changing the pump is the same as described above. All that is required is for the filter to be unclipped from the base of the machine by a small amount and the pump will simply pull out and the new one will clip in.