Sometimes a good example can instantly clarify something.

I recall back to my early school days when my math teacher explained that addition and multiplication were Commutative (a+b == b+a) and Associative ( a+(b+c) == (a+b) +c). We students didn't really understand the difference since all of the limited operations we knew were both. We asked for an example that was one but not other other. Our teacher thought and mentioned matrix multiplication. But we didn't really understand matrices and so thought that was a lame example.

Years later, it hit me that string concatenation is an operation that's associative but not commutative.  After all, "A" + ("B" + "C") == "ABC" == ("A"+"B") + "C" (associative).   But "A" + "B" == "AB", while "B"+"A" == "BA", so it's clearly not commutative.

What other common functions can you think of that are either commutative or associative?