I will assume you mean measure the "mass" or weight of air, but really the idea is similar for measuring it in any other way. In high school, you'll learn about something called the "ideal gas law" which will eventually explain this much better than this answer. But, the idea is that you can shrink or expand the air and see how its properties change. For example, if you fill a balloon with air, you can squeeze the balloon which changes its total volume (the space inside the balloon) and this also changes how much pressure is on the balloon walls (how much they stretch). By measuring how that pressure, the stretching, changes as you squeeze, and by using chemical tests to determine what the air is made of (oxygen, carbon dioxide for example), you can tell how much air there is (how many air molecules are in the balloon) and how much it would weigh, even though there is very very little of it in the space.