The integer that you find in each box of the Periodic Chart is the atomic number.
The atomic number is the number of protons in the nucleus of each atom. Notice that there is one and only one integer from 1 - 110 or so in each element box, so we have found all the elements. Each element fits neatly into its niche in the Periodic Table.
Another number that you can often find in the box with the symbol of the element is usually not an integer. It is oversimplifying only a little to say that this number is the number of protons plus the average number of neutrons in that element. The number is called the atomic weight or atomic mass.