Usually, irrationality is proven by reductio ad absurdum (a proof by contradiction). Here are some examples:

Theorem 1: √2 is irrational.

Proof:

Assume that √2 is rational and that in lowest terms, √2 = A/B.

Then we have 2=A²/B² so A²=2B².

We see that A² is even, thus A must be even. Now let A=2K and substitute.

(2K)²=2B² so 4K²=2B² so 2K²=B².

We see that B² is even, thus B must be even.

Since A and B are both even, A/B is not in lowest terms, a contradiction.

Theorem 2: log₂ 3 is irrational.

Proof:

First observe that log₂ 3 > 0

Assume that log₂ 3 = A/B where A and B are positive integers.

Then we get 2^A/B=3 so 2^A=3^B, a contradiction, because 2^A is even while 3^B is odd.

There are also proofs by contradiction that establish irrationality of e and π.

Two well known facts:

1. If an integer K>0 is not an N^th power of an integer, it's N^th root is irrational.

2. For two integers A>1 and B>1, log_B A is rational if and only if A and B are powers of the same integer:

Ex: log₄ 8 is rational because 4 and 8 are both powers of 2 (4=2² and 8=2³).

An irrational number is one that goes on and on and on and on forever and ever and ever and ever and doesn't stop and doesn't repeat.  The most common ones are pi and square roots of numbers.  

The technical definition -- a rational one is one that can be written as a fraction.  An irrational number cannot.  So remember that a number like 1/3 is 0.3333333333333333333333 (where 3 has a line over it and it goes on forever) repeats, so it isn't rational.  Some reallly smart college student tried to figure out what pi is a long time ago and he started writing it out and eventually filled up all the walls in the room he worked on with numbers.