An integer is a whole number, like 9, 63, -27, 18, -105, and so on. This does not include numbers with decimal points or fractions that don't simplify to a whole number.
So 107 is an integer, and you're asked to find an integer before and after 107. Well, since integers are whole numbers, you can figure out that the next whole number after 107 is 108, and the whole number before 107 is 106. Observe:
So 107 lies between 106 and 108.