Irrational Numbers
An Irrational Number is a number that cannot be written as a simple fraction  the decimal goes on forever without repeating.
Example: π (Pi) is an irrational number. The value of Pi is
3.1415926535897932384626433832795 (and more...)
There is no pattern to the decimals, and you cannot write down a simple fraction that equals Pi.
Values like ^{22}/_{7} = 3.1428571428571... get close but are not right.

It is called irrational because it cannot be written as a ratio (or fraction),
not because it is crazy! 
Rational vs Irrational
But if a number can be written as a simple fraction then it is called a rational number:
Example: 9.5 can be written as a simple fraction like this
^{19}/_{2} = 9.5
So it is not an irrational number (and so is a rational number)
Here are some more examples:
Number 
As a Fraction 
Rational or
Irrational? 
5 
5/1 
Rational 
1.75 
7/4 
Rational 
.001 
1/1000 
Rational 
√2
(square root of 2) 
? 
Irrational ! 
Example: Is the Square Root of 2 an Irrational Number?
My calculator says the square root of 2 is 1.4142135623730950488016887242097, but this is not the full story! It actually goes on and on, with no pattern to the numbers.
You cannot write down a simple fraction that equals the square root of 2.
So the square root of 2 is an irrational number
Famous Irrational Numbers

Pi is a famous irrational number. People have calculated Pi to over one million 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:
√3 
1.7320508075688772935274463415059 (etc) 
√99 
9.9498743710661995473447982100121 (etc) 
But √4 = 2, and √9 = 3, so not all roots are irrational. 
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 it was 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!
