These are the numbers we use for counting, starting from one, going on indefinitely as we count upwards. As well as counting we can do addition with these numbers, but as soon as we consider the opposite of addition, subtraction, we find that it doesn't always work. Multiplication is also always possible, but division only sometimes. Adding Zero makes a difference. Adding zero into the natural numbers, giving the non-negative integers, may seem today such a small step, but it was a significant development in its time.
Integers make subtraction work.
The next step is to make subtraction work properly, and to do this negative numbers are required. This is really just extending the notation to provide a notation for the solution to a subtraction where the number subtracted is larger than the original, and we call the resulting number system the integers. Now addition, subtraction and multiplication work all the time, but division doesn't. A number system is closed under an operation if the operation can be applied to any two numbers and the result is still a number. The integers are not closed under division.