SAT = site acceptance test. This is the test we use to show that the equipment does what we specified it to do. Until this test is completed successfully, we do not accept the equipment and the job is not completed. On successful completion, we "accept" the equipment and will sign the handover certificate.
FAT = Factory acceptance test. Once the machine/system/software has been assembled at the supplier's factory there should be a formal test with checkpoints and performance measures to prove that it is capable of meeting the URS (User System Requirement). NB This does not mean it has to meet the URS there and then. That may be impossible until it's finally plumbed in on site. Very often customers will ask to be present at the FAT to witness this before giving permission for the system to be shipped to their site. Some projects will have stage payments on successful completion of a witnessed FAT.