Are these two statement equivalent? $\forall x \exists y P(x,y)$

$\exists x \forall y P(x,y)$

where P(x,y) means x is smaller than y.

I believe that they mean the same thing.
Assuming you mean $\exists x \forall y P(x,y)$ for your second one, no these are not the same. 

Put them into words and it will become clearer: 

The first one says: 

" Every $x$ is smaller than some $y$. " 

The second one says: 

" There is some $x$ which is smaller than every $y$. " 

These are certainly not saying the same thing. For instance the first one is true in $\mathbb{R}$ but the second one is not.
The second condition asserts a form of uniformity.