"ɔ" Like (awesome, autumn, Australia), "ɒ" Like (octopus, October, occupy), "ɑ" Like (arm, art, argument). I know it's going to be hard to explain them in writing, but if there's a native speaker who can explain them in an voice clip, I'd be so grateful.
2/ɒ/ and /ɑ/ are the same sound for most Americans. Dictionary.com retains the distinction just out of tradition. /ɔ/ is also the same for about half of them.– NardogJun 12, 2022 at 16:44
The right procedure is to first gather sample pronunciations, then compare the vowel in those examples to a standard. You can get a collection of (36) pronunciations of "awesome" here. This includes US, UK and Australia, though bear in mind that these are random internet submissions. Then you can compare the tokens against a standard, such as the IPA expert demo page provided by the IPA, so that you know the range of pronunciations within and differences between the vowels [ɔ ɒ ɑ], as well as [ɐ a] which are in the same neighborhood. The final step, not practical, would be to submit a recording to an expert for evaluation.
user6726 has given a good answer for the actual phonetic meanings of the symbols, but since you ask specifically about their usage "in American pronunciation", I'd like to give another answer.
For me, and many other American English speakers, there is no difference. Since you're asking about these three specific sounds, I imagine you don't have this distinction either. So if you want to learn the difference, listen to recordings by trained phoneticians (as user6726 suggests), or make sure you're listening to recordings from a dialect that makes this distinction (as many English dialects do).
This is a combination of two linguistic phenomena: the "cot~caught merger" is the loss of the distinction between
/ɔ/, and the "father~bother merger" (or "Khan~con merger") is the loss of the distinction between those and
/ɑ/. Dialects that show both of these mergers, like mine, have only a single phonemic low back vowel. (These three vowels are also sometimes referred to as LOT, THOUGHT, and PALM, to describe their distribution in English without committing to any particular standard for how they should be pronounced.)
I'm from Northern New England - for me there is zero difference between the first two and a virtually negligible difference between the first two and the last.