The periodic table, element by element
Each box is one element. The table puts the elements in order of atomic number, which is the number of protons, and it stacks elements that react alike in the same column.
A worked example
Find magnesium, Mg. Its atomic number is 12, so a magnesium atom has 12 protons. A neutral atom also has 12 electrons. Magnesium sits in group 2 and period 3. Period 3 means its electrons fill 3 shells: 2, 8 and 2. Group 2 means it has 2 outer electrons. It loses them to form the ion Mg²⁺. Calcium, below it in group 2, also forms a 2+ ion. Recap: the row gives the number of shells, and the column tells you how the element reacts.
The mistake people actually make
The atomic mass is not the atomic number. Magnesium's atomic mass is 24.31, but it has 12 protons. After calcium, element 20, the shells stop filling in the simple 2, 8, 8 order. For those elements, use the electron count instead of a Bohr diagram.