I'd started with a note on potential temperature, the vertical structure of the atmosphere, gas laws, and a few other things. After much typing, I realized that I'd done much typing and hence you'd have a lot of reading to do. So I'll divide things up a bit and hope that the result is more manageable and understandable.
In the introducing the atmosphere note, I mentioned that in the lower atmosphere (troposphere) and upper atmosphere (mesosphere), temperatures decreased with height. This is true, but if you've heard about 'hot air rises', I hope it gave you some discomfort. We'll ultimately solve the problem by understanding potential temperature. But first, let's think some more about the vertical.
The troposphere is where we live, and where almost all of what we think of as 'weather' happens. It is between something like 10 and 20 km thick. Above this, is the stratosphere. It is where the ozone layer is, and temperatures are constant or increasing with height. It runs from the top of the troposphere, that 10-20 km elevation, to the base of the mesosphere (something like 50 km). The mesosphere runs from the something like 50 km to something like 80 km. Finally, above the 80 km, is the thermosphere, by which time the atmosphere is so thin that different processes take over.
The 'something like' values there should make you uncomfortable. Even more so, that 10-20 km thickness of the troposphere. Why should it vary by so much? Partly, what is happening is that in hotter parts of the atmosphere (the tropics), pressure drops more slowly with height than in cold parts (the polar regions). If we use a thickness that relates to mass (pressure) rather than temperature, we find that the troposphere is between 700 (polar) and 900 (tropics) mb thick. Still thicker in the tropics, but no longer a factor of 2, so this is progress.
Average atmospheric pressure at sea level is 1013.25 mb. This is a pressure of a column of air over you standing at that point. So the tropospheric thicknesses say that the troposphere is 70-90% of the mass of the atmosphere over a given point. If we used elevations instead, then the troposphere is only 10-20% of your vertical -- the remainder being the very thin gases of the stratosphere, and even thinner gases in the mesosphere.
The stratosphere, then, is from the 100-300 mb level (above 70-90% of the mass of the atmosphere) to about the 1 mb level. In other words, it and the troposphere jointly account for 99.9% of the mass of the atmosphere. The mesosphere runs from about 1 mb to about 0.01 mb. Add this in, and we've got 99.999 % of the atmosphere.
In terms of building models of the atmosphere, and for observing it, the pressure levels make many things easier. If we took meters (or feet) in the vertical, and spaced evenly, then we waste 80-90% of the levels on parts of the atmosphere that don't have the weather we're interested in. If we take pressure (the mb -- millibars), then we only 'waste' 10-30%, a big improvement. For observing, it is easier to build a pressure gauge than to sense the elevation above the ground. (Often, even elsewhere, pressure is used instead of elevation, such as for aviation). Radiosondes, then, report in terms of the pressure they're experiencing at the time of an observation.
If we were being strictly correct, I should be replacing references to millibars (mb) with hectopascals (hPa). The latter (well, Pascal) is the official scientific unit for pressure. 1 mb = 1 hPa, though, so we can just swap them. In other units, 1 mm mercury is a bit more than 1 mb, there being 760 mm Hg in a standard atmosphere to the 1013.25 mb. Inches of mercury ... well, my great grandmother's barometer uses that (only) but it's not even close to what's used in science and hasn't been for a long time. Then there are pounds per square inch (14.7 being standard atmosphere), and a host of others. I'm lazy about the mb versus hPa label because a) it's what I learned first b) it's in more common use in the US than hPa and c) most importantly, the conversion factor is 1. Still, if you're a younger reader, get used to the hPa and practice translating to it in your mind rather than perpetuate the erroneous mb.