Imagine you've got ten pieces you need to stick end to end

    a b c d e f g h i j

How many ways can you do that with a binary operation? i.e. how many ways can you completely parenthesise that expression. e.g.

    ((a (((b (c d)) e) ((f g) h)) i) j)

I haven't worked it out exactly, but I think it's about a trillion.

On the other hand, suppose your binary operation is associative. How many different ways are there now? Effectively one.

That's a huge reduction in complexity. The notion of associativity is extremely useful in developing composable programs.