Key definitions/explanation of terms
Ionisation energy is the energy required to remove one mole of electrons from one mole of gaseous atoms to produce one mole of gaseous ions with a 1+ charge.
Electrons carry a 1- charge and are located very close to one another in the atom. As a result of this the electrons repel one another. This effect is referred to as electron shielding. The more subshells populated in the atom the greater this shielding effect is.
The energy required to remove one valence electron from the outer shell is dependent upon the force the electron experiences due to the nucleus- this is called the effective nuclear charge.
This contributes to the ionisation energy measured as the larger the atomic radius, the further from the nucleus the valence electron is (therefore less "of a pull" experienced) so the less energy required to remove it.
When moving from left to right across a period the ionisation energy generally increases. This may seem counterintuitive to some but can easily be explained. Each successive element gains an extra proton and electron. The result of this is the size of the nucleus increases whilst the same number of subshells are being populated. As a result the valence electrons experience a greater pull from the nucleus meaning they are drawn in closer. Subsequently the atomic radius decreases and the ionisation energy increases.
A similar thought process can be applied when descending a group. Each successive element in the group populates the next set of subshells whilst simultaneously gaining protons. However, as more electrons are added the valence electrons experience a greater electron shielding from the inner subshells. Consequently, the valence electrons move further from the nucleus and less energy is required for the removal of one electron.
This can be demonstrated using the reaction of the alkali metals with water- a typical experiment which many students have witnessed. As the group is descended the reactions become more vigorous in nature. One contributing factor to this is the decreasing ionisation energy of group 1. It becomes easier to remove the valence electron as you move down the group making the element more reactive so the reaction proceeds in a quicker time frame.