Natural numbers (often denoted by N) consist of positive integers or counting numbers (1, 2, 3, ...) or, in some cases, non-negative numbers (0, 1, 2, 3, ...).  

Natural numbers have two main purposes: counting ("there are 2 dogs") and ordering ("that is the 4^th largest planet").

Integers (often denoted by Z) consist of positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...), and the number zero.  Integers are numbers that can be written without a fractional or decimal portion.  The numbers 56, 112, -13 are integers while 33.43 and 3½ are not.

A rational number is a number which can be expressed as a ratio of two integers. Non-integer rational numbers (commonly called fractions) are usually written as a/b, where b is not zero.

An irrational number is any real number that is not a rational number.  Specifically, it is a number which cannot be expressed as a fraction m/n, where m and nn non-zero.  An example of an irrational number is π ≈ 3.141592653589793...