The different types of COVID tests represent a variety of options for what you want to get tested for.
PCR tests are considered the "gold standard". PCR stands for polymerase chain reaction which is a method widely used to rapidly make millions to billions of copies of a specific DNA sample, allowing scientists to take a very small sample of DNA and amplify it to a large enough amount to study in detail. This test actually detects RNA (or genetic material) that is specific to the virus and can detect the virus within days of infection.
Antigen tests are immunoassays that detect the presence of a specific viral antigen, which implies current viral infection. Unlike nucleic acid based tests such as PCR, which detect the presence of genetic material, antigen tests detect proteins, such as those found on the surface of a virus. Accuracy can be a problem, with antigen tests typically having a much lower sensitivity than PCR. (amendment, We highly recommend receiving a PCR molecular test alongside antigen/antibody tests to additionally verify the results for extreme precision).
An antibody test reveals if a person has already been exposed to an infection, by detecting antibodies in their blood or serum. Antibody tests are not usually used to diagnose current infection as it takes the body some time to produce antibodies. A negative test result may occur if the test is taken too soon after infection before antibodies have been produced by the body.
For more information on the differences between PCR, Antigen, and Antibody tests view this DC Health Document