An irrational number is any real number that is not a rational number. A rational number can be defined in the form a / b (i.e., a divided by b) where a and b are integers. So in other words, an irrational number is a number that cannot be expressed as a fraction of two integers.
5/3 of all people do not understand fractions.
Another way of looking at it is to say an irrational number is a number whose decimal representation neither repeats nor terminates. While a mathematician may argue the validity of this alternate definition1, it is correct enough to help you understand irrationals if you don't quite understand the definition in the first paragraph.
Let's look at some examples of rational numbers using the second definition:
Let N = .125
N does terminate (i.e., the decimal does not continue on forever), and it can be represented by a fraction - in this case 1/8, so .125 is a rational number
Let N = .381818181818181... (the "81" repeats infinitely)
Since N does repeat, it too is a rational number. To prove this, let's convert it to a fraction.
N = .381818181818181...
Since there are 2 numbers that repeat, multiply both sides by 10^2 (100)
100N = 38.1818181818181...
Subtract N from both sides
99N = 37.8
Divide both sides by 99 and reduce
37.8 378 21
N = ---- = --- = --
99 990 55
Whether the repeating part is a single number (0.333333...) or a billion numbers, the process is exactly the same other than the number you multiply by in the second step.
As displayed above, even if a number repeats off into infinity, it still can be rational. If this is the case, what would an irrational number look like? Unfortunately, because they never terminate or repeat, there is no way to actually type out an irrational number. Because of this, they are often given names or written out in mathematical formulas:
While it has been proven that the examples above are irrational, it can be an extremely difficult process to create a mathematical proof showing that a number is irrational. You can't just look at it and determine one way of the other. Let's say you have the first 1000 decimal places of a number, and there are no repeating patterns. You cannot just assume that it is irrational because it is possible that those 1000 decimal places are the first part of the repeating pattern (i.e., the second 1000 decimal places are exactly like the first). You can continue extending this example out to one million, one billion, etc, and you will never know just by looking. If you are interested in how some numbers have been confirmed to be irrational, see the related links at the bottom of the writeup.
If you are still confused about irrational numbers, don't be too concerned. Even Pythagoras had trouble with them at first. It is said that one of his aids, Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. Unfortunately, Pythagoras still believe that all numbers were absolute (though he could not prove it), and would not accept the existence of these so-called irrationals, and had Hippasus thrown overboard (where he drowned) for his ideas. Pythagoras later wrote the first proof of their existence.
More attention was given to these numbers in the 3rd century BC by Euclid in book 10 of Euclid's Elements. Very little study was given to irrationals from this time until the late 1700s and 1800s where Johann Heinrich Lambert, Paolo Ruffini, Karl Weierstrass, Heine, Georg Cantor, Richard Dedekind, and numerous other mathematicians all studied and wrote about them. Today we learn about various irrational numbers in high school, and some people use the more common ones (pi, e, golden ratio) every day in such fields as engineering, physics, mathematics, architecture, or computer science.
1 For those of you who asked, the second definition does not hold true for all bases. For example, in base π, the second definition would consider π to be rational (since it equaled 1). If you are working strictly in base 10, the second definition should alway be true.