What does an irrational number mean?

An irrational number is a (real) number which is not rational, this is the definition. A rational number, by the definition, may be expressed as m/n, where m is an integer and n is a natural number. Hence an irrational number may not be expressed this way.
Some irrational numbers occur naturally from Pythagorean theorem, for example if a right triangle has both legs of length 1, then the length of its hypotenuse is sqrt(2), an irrational number (ask me for proof if needed).
Written in decimal form, an irrational number has infinitely many digits after the decimal dot, and there is no period in them. This is also in contrast with rational numbers.
From the set theory point of view, there are much more irrational numbers than rational: the set of rational numbers is countable and the set of irrational numbers has the cardinality of continuum.
Despite of this, there are enough rational numbers to approximate any irrational number with any accuracy. In other words, the set of rational numbers is dense everywhere in the set of real numbers.