There are a number of major sets of numbers.

Natural Numbers.

Firstly, there are the natural numbers or counting numbers. These begin with either one or zero, and continue upwards with all whole numbers greater than one or zero (depending on the definition). There are infinitely many.
Examples: (?), 1, 2, 3, 4, 5, 42, 69, 666, 1701, 2600.

Integers.

These are any 'whole numbers', including negative numbers, natural numbers, and zero if not counted already.
Examples: (all above plus) -1, -2, -5, -999.

Rational numbers.

These are any numbers which can be written as
p/q
where p and q are integers. p and q are the usual letters used. This therefore includes all types of fractions, whether proper or not, all decimals (including those which are recurring). There are infinitely many rational numbers between any two different integers.
Examples: (all above plus) 0.2, 4/5, 66/9, -12.2.

Real Numbers

These are all numbers which can exist from performing any form of arithmetic operation on any set of real numbers. These include irrational numbers, which include pi, e, and the square root of any natural number which is not a perfect square. There are infinitely many of these between any two different rational numbers.
Examples: (all above plus) pi, e, sqrt(12.21).

Complex Numbers

These are all known numbers, which are used to calculate the square root of -1, which is defined as i. With this, a whole new set of mathematics occurred, including fractals. Complex numbers that are not real are imaginary.
Examples: (all above plus) i, 3i+4.

Venn diagrams.

(((((A: natural)B: integer)C: rational)D: real)E: complex)
anything in B but not A is negative.
anything in C but not B is a fraction.
anything in D but not C is irrational.
anything in E but not D is imaginary.

Help wanted: /msg me!