'아무' means 'any' or 'any old' something or someone. When referring to something, the thing will immediately follow '아무'. When referring to someone, you can use '아무' without adding a word for 'person'. In a negative context, it would mean 'none (at all)'.
The noun that follows '아무' will have either the particle '-(이)나' attached to reinforce this meaning of 'one but none in particular' or, if it's being used in a negative context, it will have the particle '-도' attached and the main verb will also be negative. I'm trying to think of any counterexamples to this but can't come up with any off the top of my head.
For example, 아무 때나 좋아 => anytime is fine. (You cannot say 아무 때가 좋아.)
'아무도' means 'nobody'. For example, 집에 아무도 없다 => there is no one at home. Likewise, you cannot say 아무가 없다.
If you wanted to say "Nobody knows" then it would be "아무도 모른다" (literally, "Even any old person doesn't know"). Note how the verb is negative.
'아무것도' means 'nothing'. For example, 집에 아무것도 없다 => there is nothing at home. You cannot say 집에 아무것이 없다.
Contrast this with '아무것이나' (or more commonly, '아무거나), which means 'any (old) thing':
Q: 뭐 먹고 싶어? What do you wanna eat?
A: 아무거나 좋아. Anything is good.
or... 아무것도 먹고 싶지 않아. I don't want to eat anything.
There are lots of other expressions based on '아무', as well, but they all have that basic sense of 'any'.
In casual speech, 아무 can be abbreviated to '암'.