There are many (greater than 20,000) kinds of psychological tests. Most are not used, or are only used in vary specific and limited situations.
Schools often employ masters level psychologists to administer, score, and interpret those psychological tests that the school uses to evaluate learning disabilities and matters related to being able to learn. Public Schools are required by law to test a student when the parents ask, in writing, for testing. Sometimes problems are found that do not meet the threshold requirements under which the school must offer to help with learning, however other avenues of help are often available to parents who know their child has some areas of struggle.
Psychological testing is sometimes used to screen for some symptoms or diagnoses, like suicidal thoughts or depression. Sometimes certain psychological tests are helpful in figuring out or understanding the mind when the usual methods of understanding aren't working well enough.
Psychologists are usually the people administering these tests because they have received specialized training in their schooling. Sometimes others have been specially trained to administer a test, or sometimes even score and interpret a test. The most important point here is training. Psychologists have specialized training in the course of getting their degrees, and may have a specialized interest with further additional training.