The successive ionisation of an element is the removal of its electrons one at a time. For example, for the element boron (atomic number 5) the following equations represent the process of successive ionisation.
B(g)B(g)+ + e-
B+(g)B(g)2+ + e-
B2+(g)B(g)3+ + e-
B3+(g)B(g)4+ + e-
B4+(g)B(g)5+ + e-
The first equation starts with an isolated boron atom in the gas state and finishes with a singly charged boron cation also in the gas state and an electron.
The second equation starts with a singly charged boron cation in the gas state and finished with a doubly charged boron cation in the gas state and another electron.
The third equations starts with a doubly charged boron cation in the gas state ... and so forth.
The energies required to bring about the changes shown above are referred to as successive ionisation energies.