a rational number is what you know as a fraction. It is the ratio of two whole numbers.
3/4, -7/11, etc.
An integer is a rational number with a denominator of 1. Or, more commonly just what you think of as a counting number, but may be negative:
3/1 = 3
-2/1 = -2
and so on
An irrational number cannot be expressed as a fraction. When written down as a decimal, they never terminate, so a decimal value is not exact. There are various kinds of irrational numbers, such as
√2 = 1.414...
e = 2.71828...
π = 3.14159...
∛17 = 2.57128...
The special symbols used on the left give their exact values. The decimal equivalents are just an approximation.