A manometer is a device that measures the pressure of a fluid. It is comprised of a tube filled with a liquid, and the greater the pressure of the fluid being measured, the higher the liquid in the manometer gets pushed up. I've attached a simple picture of its mechanism below.
Scientifically, a fluid is gasses and/or liquids. Therefore, a manometer can be used to measure air pressure. Air pressure works just like any other fluid. The deeper that you go within that fluid, the greater the pressure. If you have ever dived down to the bottom of an 8 foot deep swimming pool, you have experienced the increase in pressure that coincides depth increase. This is also why you feel pressure changes on your ears whenever you have ridden in a car going up or down a mountain road.
Think of the air/atmosphere like a really deep swimming pool. The atmosphere is generally considered to be 100 km (60 mi) deep. At sea level, you are just about as deep as you can go in the atmosphere (Death Valley is actually below sea level). The manometer would register a pressure. If measuring in Pascals it would be 101 kPa. As you gain elevation, air pressure would begin to decrease, and the manometer's fluid would drop down to indicate a pressure decrease. Air pressure and elevation are inversely related. As elevation increases, pressure decreases and vice versa.