Types of numbers
In a nutshell
Numbers are used to represent quantities of items, and different types of numbers belong to different sets. For example, integers can be used for counting while rational numbers are better used to describe how much of a portion remains.
Identifying types of numbers
Type
 Description
 Example

Natural
 Any positive whole number is a natural number (not including $0$)
 $1, 2,5$

Integer  Any whole number is an integer (including $0$)
 $0,7, 3$

Rational  Any number that can be expressed as a fraction where in its simplest form both the numerator and denominator are integers
 $\dfrac{16}{7}, \dfrac{1}{3}, 9$

Irrational  Any number that cannot be expressed as a fraction
 $\pi, \sqrt{2}, \sqrt{\dfrac{17}{3}}$

Real  Any number is a real number
 $\sqrt{3}, 4, \dfrac{3}{5}$

Prime  Any positive whole number that has only two factors, that being $1$ and itself
 $2,5,13$

Square  Any number that can be expressed as the square of a natural number
 $1, 4, 16$

Cube  Any number that can be expressed as the cube of a natural number
 $1, 8,27$

Surd  Square rooted numbers and can be positive or negative
 $\sqrt{50},\, 3\sqrt46,\, \sqrt{8}$

Example 1
What type of number is $8.6$?
$8.6 = \dfrac{86}{10} = \dfrac{43}{5}$
Hence as this is the simplest form and both the numerator and denominator are integers, $8.6$ is a rational number.
Subsets of numbers
Some of the definitions of numbers overlap. For instance, $6$ falls into the category of a natural number, integer, rational number and also a real number. This gives a subset order for the types of numbers which can be seen below:
From the Venn diagram you can see that all natural numbers are integers, all integers are rational numbers, and all rational and irrational numbers are real.