Validity, to give a short and non-technical definition, means “the extent to which a test measures exactly what the test maker intends it to measure, nothing more, nothing less.” (definition courtesy of Donald E. Powers, ETS).
Why is understanding a test's validity important? It's simple--not knowing how valid a test is means you don't know to what extent results are meaningful and actually provide the information you need.
How can you determine how valid a test is? As a test user, you can simply ask the testing company for test development background and validity statistics. For the test's creator, however, ensuring validity is a huge challenge.