pronunciation - What exactly is the "schwa" sound?

What exactly is the "schwa" sound? As a non-native speaker, I hear this sound as not being a pure and clean sound. I mean I know that every vowel sound may vary depending on whether the syllable is stressed, on the accent of the person that makes the sound, etc. But generally this sound is the same in the sense that it does not depend on the consonant sounds that come before or after, and might or might not be heard as being different by the non-native speaker ear.

The schwa sound is a very difficult one for me because I cannot find a pattern to follow. When I was learning the other vowel sounds I could analyze a long list of words being pronounced (by the same person of course) and then abstract the sound so I can produce it perfectly. But this does not work for the schwa sound. So for example I hear one sound in words like a-bout, b-a-loon, decim-a-l, and a different sound in words like s-u-ppose and impet-u-s, t-o-day or t-o-night. Also in words like Ros-a-'s and ros-e-s, the schwa sounds differently.

When it comes to words that have the schwa sound on vowels e an i like in the word insan-i-ty then the sound is almost the one for the i in t-i-p, but also in words like b-e-hind I hear some people say it with a schwa sound like in a-bout and some other people pronounce it like the i in p-i-t. So, I would really appreciate an explanation about how you perceive this sound and how you would explain it to someone who, like me, is not a native speaker.