What's $P$ and what's $Q$ in this classic proof of the irrationality of $\sqrt 2$? In this proof extracted from the Wikipedia


  A classic proof by contradiction from mathematics is the proof that
  the square root of 2 is irrational. If it were rational, it could
  be expressed as a fraction $a/b$ in lowest terms, where a and b are
  integers, at least one of which is odd. But if $a/b = \sqrt 2$, then $a^2=$
  $2b^2$. Therefore $a^2$ must be even. Because the square of an odd number is
  odd, that in turn implies that $a$ is even. This means that $b$ must be
  odd because $a/b$ is in lowest terms. On the other hand, if $a$ is even,
  then $a^2$ is a multiple of $4$. If $a^2$ is a multiple of $4$ and $a^2=2b^2$,
  then $2b^2$ is a multiple of $4$, and therefore $b^2$ is even, and so is $b$. So
  $b$ is odd and even, a contradiction. Therefore the initial
  assumption—that $\sqrt 2$ can be expressed as a fraction—must be false.


Knowing that a proof by contradiction you assume P and Not(Q) what's P and what's not Q in this proof?
I'm assuming you are more accustomed to seeing proof by contradiction used largely with statements that are implications or conditionals. And indeed, when writing a proof by contradiction to prove statements of the form $$P \implies Q,$$ we typically assume $(P\land \lnot Q)$.

But in this particular case, we do not seem to have an implication to prove. Rather, we have the proposition: 


  The square root of $2$ is irrational. $\quad$( $Q$). 


There's no helpful "if, then", or "this implies that" to indicate any sort of implication being asserted. So we have an example of the use of a proof by contradiction where to prove a statement other than an implication.

What we can do is to think of the assertion to be proven as a simple "atomic" proposition: $\,Q.\,$ Then $\,\lnot Q\,$ is the statement to the effect: 


  Suppose $\,\sqrt 2\,$ is not irrational. $\;$ Put differently, suppose $\,\sqrt 2\,$ is rational.$\quad(\lnot Q)$


The proof then proceeds, after having supposed $\,\lnot Q\,$ to invoke the definition of a rational number in order to arrive at a contradiction.

In a sense then, the proof amounts to a "bare-bones" proof-by-contradiction: 

To prove that $\,Q,\,$ we assume $\,\lnot Q,\,$ and then we work to obtain a contradiction. Once we arrive at a contradiction, we can conclude that our assumption is false, and so we are justified in negating the false assumption: "therefore, $\lnot\lnot Q.$" $\;\;$ And this amounts to affirming the desired conclusion/assertion: therefore $Q$, since $\;\lnot \lnot  Q\equiv Q$. 

The contradiction in this proof happens to come from our knowledge about the rational numbers, information which could be considered a premise: the "implicit" premise $P$ being the definition of a rational number.
P would be that $\sqrt{2}$ is rational, Q that it is irrational, and so not(Q) is that it is not irrational.