Quantifier Notation

What's the difference between $\forall \space x \space \exists \space y$ and $\exists \space y \space \forall \space x$ ? I don't believe they mean the same thing even though the quantifiers are attached to the same variable, but I'm having a hard time understanding the difference. Any examples to make the distinction clear would be appreciated.

Solution 1:

Consider this example.

For all $x\neq 0$ there is a $y\neq 0$ such that $xy = 1$.

There is a $y\neq 0$ such that for all $x\neq 0$ you have $xy = 1$.

You can probably see that the one statement is true and the other false.

Solution 2:

Every natural number has a successor. There is no natural number which is the successor of every number.