The proof of the irrationality of root 2 is often attributed to Hippasus of Metapontum, a member of the Pythagorean cult. He is said to have been murdered for his discovery (though historical evidence is rather murky) as the Pythagoreans didn't like the idea of irrational numbers.

This means that $a^2$ is an even number: it's a multiple of $2$. Now if $a^2$ is an even number, then so is $a$ (you can check for yourself that the square of an odd number is odd). This means that $a$ can be written as $2c$ for some other whole number $c$. Therefore, $$2b^2=a^2=(2c{)}^2=4c^2.$$ Dividing through by $2$ gives $$b^2=2c^2.$$ This means that $b^2$ is even, which again means that $b$ is even. But then, both $a$ and $b$ are even, which contradicts the assumption that they contain no common factor: if they are both even, then they have a common factor of $2$. This contradiction implies that our original assumption, that $\sqrt{2}$ can be written as a fraction $a/b$ must be false. Therefore, $\sqrt{2}$ is irrational.

This proof is famous example of something called a proof by contradiction.  You can read more about this type of proof in the article Something from nothing?