The second ionization energy is the energy required to remove the electron after a one has been removed.

The second ionization energy is the energy required to remove the second electron after a valence one has been removed.

For an element, the first ionization energy is defined as the amount of energy required to remove one electron from the outermost valence shell of a neutral atom. Removing one electron increases the number of protons, making it a 1+ ion.  

The nucleus (protons) has more bonding to the electrons with negative charge and thus more energy is required if another electron needs to be removed. This higher energy required to remove second electron from a 1+ ion (after the first one has been removed) is termed as the second ionization energy. Second ionization energy leads to formation of a 2+ ion. Similarly, third ionization energy is higher than second ionization energy.

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