PAT Testing stands for Portable Appliance Testing, people normally say PAT testing for short(even though that would stand for portable appliance testing testing).

Workplaces and suppliers of electrical equipment have a duty of care to ensure the appliances are safe to use.
PAT testing is one part of the process of showing reasonable steps are made to comply with safety legislation.

The process involves a careful visual examination of the item, this is where the majority of problems are identified.
If the formal visual inspection is passed then where required a series of further more specialist electrical tests are run to test the safety of the equipment.
In addition, where possible the PAT test process identifies issues related to safe usage of the equipment (ie is it used in an appropriate location, can it be safely disconnected in an emergency as well as issues such as overloading sockets and extension leads etc.)

Once it has been identified that the item is safe to use in its location and environment, and the visual inspection (where the cable, plug, fuse, appliance itself are checked for damage/compliance with relevant safety standards) is passed and any required electrical tests are completed with safe limits the appliance is deemed safe to use at that time and a pass sticker is applied.

Appliance that fail any of the tests will be labelled with a fail sticker and removed from service and put into quarantine, the responsible duty holder is informed and they can either be repaired or disposed of safely.