What does the atomic number of an element mean?

The number of protons or electrons an atom has.
is organised by the . If you look at it, you will see it begin at hydrogen with ##1##, then ##2##, helium, ##3##, lithium, ##4##, beryllium, and so on. The atomic number is what number atom it is.
In a scientific sense, the atomic number is the number of protons you find in the nucleus of the atom, or the number of electrons floating around the outside. Protons and electrons come in the same number in atoms. (Ions are where the electron number differs from protons.)
For example, in lithium, the atomic number is ##3##, which means it has ##3## protons in the nucleus and ##3## electrons around the outside. In ionic lithium, it loses an electron and ends up with only ##2## electrons, though the number of protons remains the same at ##3##.