Integers are an important part of calculations in many different branches of thought, ranging from banking to sports to weather. Playing a video game, reviewing deposits or withdrawals in a checking account or even looking at weight all require integers. They not only show a direct relationship to some starting point, but they also give description and meaning to the numbers that occur in everyday situations.
Integers include the natural numbers (1, 2, 3, ...), the number "0" (zero) and the opposites of the natural numbers (-1, -2, -3, ...), often referred to as negative numbers. Fractions and numbers involving decimals are not integers; further, square roots that do not result in a natural number are also excluded.
Without the ability to use the negative numbers, weather would likely only be referred to as "cold" without any specific meaning about how cold (or perhaps dangerous) the weather is. Integers give a better description, allowing for the difference between -5C and -50C to be understood.
In financial work such as accounting or banking, a lack of negative numbers would require moving large amounts of currency by hand as the only expression of change. Integers as a whole complete the descriptions required for deposits and withdrawals, and a system of accounting has grown around this to describe transactions precisely and accurately.
A world without integers would be a world severely limited in description and sophistication.