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.
 A: 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. 
A: The second condition asserts a form of uniformity.
A: For some relations $P$
$\forall x\exists y P(x,y)$  (1)
is true while 
$\exists y\forall x P(x,y)$  (2)
is false. So, (2) does not necessarily follow from (1). 
Example
Let the domain of quantification be $U=\{x,y\}$ for distinct $x$ and $y$, and let $P$ be the "is equal to" relation on $U$. It can then be formally proven, by examining each case, that (1) is true (every element of $U$ is equal to itself) while (2) is false (no element of $U$ is equal to every element of $U$).
See formal proof (using the DC Proof 2.0 format) at http://dcproof.com/PopSci.htm 
EDIT:
For a slightly different approach see:
$U=\{ 0,1 \}$ and $P(a,b)\leftrightarrow a=b$ at http://dcproof.com/USet-0-1.htm 
$U=N$ and  $P(a,b)\leftrightarrow b>a$ at http://dcproof.com/GTonN.htm 
