Integers and rational numbers are part of the set of real numbers. Integers are any whole number, positive or negative, on a number line. An example of some of these might be -2, 3, 10, or any other whole number. Rational numbers include integers, but include numbers such as 5/8 (a fraction), or 2.3 (a decimal representation).
Irrational and transcendental numbers are also real numbers. Irrational numbers are numbers such as the square root of 2. This number cannot be represented as a fraction of any two other numbers, and as such is considered "irrational." Irrational numbers that have no sequence of algebraic operations (powers, roots, sums, multiplications) that will add to its value are "transcendental." Examples of these would be pi and e.
There are two primary and easily understood mathematical concepts that aren't real numbers. Complex numbers include real and imaginary numbers. Imaginary numbers are any number with a square that is negative. These are not real numbers, and as complex numbers are partially comprised of these, they are not real as well. Infinity, which most commonly describes an unbounded limit for x either growing or decreasing, is also not a real number. However, it is not a number in the conventional sense of the term at all.
The simple definition of real numbers as values along a number line is useful, but not at the higher levels of mathematics. For higher levels of mathematics, real numbers are complete totally ordered fields. However, this definition is only needed for high-level calculus, numerical analysis and set theory.