Atmospheric pressure, put quite simply, is the power that air molecules exert against a surface. Contrary to what we may see and feel, 'air' does have mass, meaning it is comprised of different molecules, and those molecules feel the need to constantly move randomly around and put pressure on other molecules around them. In other words, air wants to occupy the most space possible, and so it exerts pressure. The only reason why most objects don't become completely squashed is because those objects have molecules of their own that push back against the air, balancing out the pressure.
Atmospheric pressure is much more dense at sea-level, and becomes less 'stuffy' the higher up in the atmosphere. The weight of air pushes down on the air underneath it, and so, a lot of air can be considered as trapped between air and a hard place. The higher up you go, the more scattered the air is, which is why when climbing a mountain you will notice that breathing becomes harder. There is simply less air to breathe.
Atmospheric pressure is actually a unit of pressure, calculated in Inches of Mercury, or Millibars, as well as a few less used units.
Inches of Mercury is a column of mercury marked off with inch markings. The normal atmospheric pressure at sea level is 29.92 inches. Many weather forecasts use this method of measuring. It is better known as a barometer, although barometers do not necessarily use mercury. Most use water in place of the volatile liquid. A barometer is a little reservoir of liquid, be it water of mercury, with a pipe-like column standing in the middle. The column is closed at the top, and when atmospheric pressure pushes down on the open reservoir, the liquid rises in the tube, and using measurement marks, the pressure can be calculated.
Millibars are the same as bars, but at one thousandth of them. A bar, from the Greek 'baros' meaning weight, is a general unit of pressure used to express the amount of force it takes to move an object weighing one gram, being one centimeter, in one second. The normal atmospheric pressure at sea level is 1013.2 millibars. Millibars are used on weather indicating maps, mostly.
When forecasting weather, low atmospheric pressure, i.e. when the air is too scattered and therefore not heavy enough to push down on more air, usually means that the sky will be clear and the weather agreeable. On the other hand, high atmospheric pressure, i.e. when the air is very dense and is even heavier than normal, trapping a lot more air at ground level than usual, indicates bad weather, such as precipitation and storms. Pressure changes according to the density of air molecules, which in turn changes in accordance to temperature. When molecules get warmed up, they vibrate more violently, thus pushing other molecules away more efficiently. In the end, the air molecules end up very spaced out and scattered, giving us low atmospheric pressure. The exact opposite is true. When the molecules cool down, their pressure is constricted and thus they end up packed close together, resulting in high atmospheric pressure. Atmospheric pressure changes constantly, but it does so so minutely that the changes are not observable.