One of the main uses of integers is to add and subtract finite quantities. For example, if you are running a market and wanted to know the total number of products you sold, you would find the number of each product in your market and add them together. In this example, you would be adding integers to other integers to produce a final integer. Another common example of using integers to add quantities is age. Each year, you add one more integer to your total age.
While fractions are not themselves integers -- hence the term "whole numbers" -- they are always made up of integers. More formally, any fraction can be written as p/q, where p and q are integers, provided that q is not zero because it is impossible to divide by zero. For example, imagine a pie with 12 slices. If 9 slices have been eaten, the total fraction of the pie left is 3/12 = 1/4 = 25 percent. The total percentage remaining was found by dividing integers.
In several branches of mathematics, you will often find expressions like f(x) = x^n, where n is an integer. In other words, this expression is representing abstractly all the possible functions f(x) = x, f(x) = x^1, f(x) = x^2, etc. It is important for these types of functions that n cannot be a fraction. In calculus, for example, the technique for finding the derivative of a function f(x) = x^n will depend on whether or not n is an integer.
Integers play a significant role in everyday finance and accounting. For example, if you have a balance of $100 at a bank and you debit $40 at the grocery store, that will show up on your bank balance as -$40, and the original $100 will shrink to $60. The practice of accounting often consists of making sure that the column of negative numbers remains in balance with the column of positive numbers -- leading to the phrase "balancing the books."