An integer is a number that can be written without a decimal or fractional component. The set of integers is a subset of the real numbers composed of the natural numbers (1,2,3...), zero, and the negatives of the natural numbers (-1,-2,-3...). This set of numbers is often denoted by a boldbace Z.
Of the numbers above, 1, -2 and -25 are integers. The number 6.78 is NOT an integer because it cannot be written without its decimal component.