You use an integer when you tell your age! Anytime you count anything, you are using integers. That’s because integers are all the positive and negative whole numbers, including the natural numbers you first learned. So you’ve been using integers all your life.
Perhaps you really want to know, how are negative numbers used (since you were probably introduced to the term “integer” as a way of extending the natural numbers to include negative numbers)?
For people who stay close to the Himalayas, where the climate is extremely cold, they use negative numbers (often negative integers) to tell the temperature in the winter, say -10 deg Celsius or even -30 deg Celsius.
Scientists or Engineers, will find negative numbers essential. Even when quantities we measure, like mass or pressure, are always positive, we need negative numbers to describe changes or differences or rates of change, because quantities can both increase and decrease.
Money also can increase and decrease, so you need negative numbers in finance–though in that world they are only slowly catching up with the modern world and learning to write “-10” instead of “(10)” or use “red ink” to indicate negative numbers!