Rational numbers are those which can be written the form a/b, where a and b are integers. In other words, they are numbers that are can be written as a ratio of two integers.
Unlike integers, the set of rational numbers (denoted by Q; think "quotient") is closed with respect to the four basic arithmetic operations (as long as the denominator of a division is not zero).
Can any operations take us out of the realm of rational numbers? What if you take (say) the square root of 2 (or any number which is not a square)? Is that rational or not? Pythagoras and the Pythagoreans originally believed that any length could be represented as the ratio of two natural numbers. Around 500 B.C., however, Hippasus of Metapontum showed that the square root of two is an irrational number; in other words, one that is not the ratio of two integers. Later on, Theatetus (417–369 B.C.) proved that the square roots of integers that aren't perfect squares are all irrational.