We (as humans) probably first used numbers in order to keep track of our sheep or count our coins. All we needed was 1, 2, 3, and so on. We now call these counting or natural numbers.
You can add and multiply counting numbers and the results will still be counting numbers. However, once you start doing subtraction, you find taking 7 away from 7 gives you zero, which is not a counting number. And if you take 5 away from 3 you get -2, which introduces the concept of negative numbers.
Integers are the set of numbers that include the counting
numbers, zero and negative numbers. Integers are quite useful until you include division, as when we divide 7 into 10 equal parts. Here you get what are called fractions, in this case 7/10.
But, of course, these are just a few of the types of numbers you might come across. There are circumstances where numbers cannot be represented by integers or fractions. Two such examples are the square root of 2 and pi, which is the ratio between a circle's diameter and its circumference). These are called irrational numbers.
When you take the set of all integers, fractions, and irrational numbers you have real numbers.
There are also imaginary numbers, which is what you get when you multiply a real number by the seemingly impossible (represented by the symbol i). When you add a real number to an imaginary number you get a complex number.
I learned all this on Sesame Street. Thanks, Ern!