You can use the absolute value function to measure distance. For example, suppose you travel 10 miles east and then reverse and travel 10 miles west. There are two ways to think about the distance traveled. Looked at one way, you have traveled no distance at all: You are right where you started. Looked at another way, you have traveled 20 miles. The solution is to look at distance on a line as a signed value. So 10 miles west = +10 and ten miles east = -10. Then, for the first view of distance you add the absolute values |10| + |-10| = 10 + 10 = 20. For the other, you add the values themselves: 10 + (-10) = 10 -10 = 0.
It is often important to know if you are very far from average. For example, if you are much taller or much shorter than the average person, then you will have trouble with furniture and other things designed for average size people. To see how different you are from the average person, you could take the absolute value of the difference between your height and the average height of people.
In statistics, it is useful to have a measure of how spread out a group of numbers is. For example, suppose a group of five people has incomes of $10,000, $20,000, $30,000, $40,000 and $50,00. One way of measuring this is with the median absolute deviation. To get this for a group, find the median and then look at the differences between each score and the median. Take the absolute values and then find the median of those. For example, suppose a group of five people has incomes of $10,000, $20,000, $30,000, $40,000 and $50,000. The median is $30,000, the deviations are -$20,000, -$10,000, $0, $10,000 and $20,000. The absolute values of those are $20,000, $10,000, 0, $10,000 and $20,000 and the median of those is $10,000.
When making a product, it is often important that it be close to a certain size, but not important if it is slightly too small or slightly too big. For example, if a company made ball bearings, they might them to 1 inch in diameter, but they might accept anything 1/16 inch either side of 1 inch. They can take the absolute value of the difference in size between a particular ball bearing and the ideal of 1 inch. If this is under 1/16", they can accept it.