The psychometric properties of a psychological test relate to the data that has been collected on the test to determine how well it measures the construct of interest.
In order to develop a good psychological test, the new test is subjected to statistical analyses to ensure that it has good psychometric properties.
There are two broad types of psychometric properties that a test must have in order to be considered a good measure of a particular construct. The first type is called "reliability." This is the test's ability to measure the construct of interest consistently.
For example, if someone takes a test to determine their likelihood of having borderline personality disorder (BPD), their results on that test should be very similar if they take the test today and again in six months. This is called "test-retest reliability." Tests should also demonstrate other types of reliability, including internal consistency (i.e., all the items in the test should be measuring the same construct).
The second broad property that a good test has is called "validity." Validity refers to how well the test accurately measures the construct of interest. For example, the results of a good borderline personality test should be highly related to behavior that is typical in borderline personality disorder (for example, someone with a high score on a BPD test should also have a lot of problems with emotion regulation). There are also many different kinds of validity that can be assessed using statistics.