Irrational Numbers
An Irrational Number is a real number that cannot be written as a simple fraction.
Irrational means not Rational
Examples:
Rational Numbers
OK. A Rational Number can be written as a Ratio of two integers (ie a simple fraction).
Example: 1.5 is rational, because it can be written as the ratio 3/2
Example: 7 is rational, because it can be written as the ratio 7/1
Example 0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3
Irrational Numbers
But some numbers cannot be written as a ratio of two integers ...
...they are called Irrational Numbers.
It is irrational because it cannot be written as a ratio (or fraction), not because it is crazy! 
Example: π (Pi) is a famous irrational number.
π = 3.1415926535897932384626433832795 (and more...) You cannot write down a simple fraction that equals Pi. 
The popular approximation of ^{22}/_{7} = 3.1428571428571... is close but not accurate.
Another clue is that the decimal goes on forever without repeating.
Rational vs Irrational
So you can tell if it is Rational or Irrational by trying to write the number as a simple fraction.
Example: 9.5 can be written as a simple fraction like this:
9.5 = ^{19}/_{2}
So it is a rational number (and so is not irrational)
Here are some more examples:
Number  As a Fraction  Rational or Irrational? 

1.75  7/4  Rational 
.001  1/1000  Rational 
√2 (square root of 2) 
?  Irrational ! 
Square Root of 2
Let's look at the square root of 2 more closely.
If you draw a square of size "1", what is the distance across the diagonal? 
The answer is the square root of 2, which is 1.4142135623730950...(etc)
But it is not a number like 3, or fivethirds, or anything like that ...
... in fact you cannot write the square root of 2 using a ratio of two numbers
... I explain why on the Is It Irrational? page,
... and so we know it is an irrational number
Famous Irrational Numbers
Pi is a famous irrational number. People have calculated Pi to over a quadrillion decimal places and still there is no pattern. The first few digits look like this: 3.1415926535897932384626433832795 (and more ...) 

The number e (Euler's Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this: 2.7182818284590452353602874713527 (and more ...) 

The Golden Ratio is an irrational number. The first few digits look like this: 1.61803398874989484820... (and more ...) 

Many square roots, cube roots, etc are also irrational numbers. Examples:


But √4 = 2 (rational), and √9 = 3 (rational) ... ... so not all roots are irrational. 
Note on Multiplying Irrational Numbers
Have a look at this:
 π × π = π^{2} is irrational
 But √2 × √2 = 2 is rational
So be careful ... multiplying irrational numbers might result in a rational number!
History of Irrational Numbers
Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to represent the square root of 2 as a fraction (using geometry, it is thought). Instead he proved you couldn't write the square root of 2 as a fraction and so it was irrational.
However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned!