What determines the energy of an electron?

The orbital in which it exists - Electrons in orbitals with a higher quantum number have higher energies
The nuclear charge of the atom - Electrons in atoms with a higher nuclear charge have lower energies
Other electrons present - Electrons in the same or lower shells repel each other, and in doing so, shield the electron from the attractive force of the nucleus, increasing the energy of the electron

Nuclear charge: Number of protons in the nucleus

Shielding effect: An electron can be repelled by electrons in the same or lower orbitals, thus decreasing the nuclear charge experienced by the electron in question.

Trends in Ionisation Energies across a period

In general, ionisation energies increase moving across a period.

Moving across the period, there is an increase in nuclear charge as more protons are added to the nucleus. This increases the attraction to the nucleus felt by each electron, decreasing its energy.

More electrons are added moving across the period. This increases the electron-electron repulsion and shields the electron from the attractive force of the nucleus. This increases the energy of the electron.

However, the increase in nuclear charge is more significant than the increased shielding effect. So there is a general decrease in electron energies moving across the table, and thus an increase in ionisation energies.

Trends in Ionisation Energies down a group

In general, ionisation energies decrease moving down a group.

Moving down the group, the nuclear charge increases as more protons are added to the nucleus. This increases the attractive force of the nucleus felt by each electron, decreasing its energy.

However, each electron is added to a new quantum shell. Each quantum shell is higher in energy and electrons in higher shells also experience more repulsion from electrons in lower shells. These two factors lead to an increase in the energy of the electron.

However, the effect from the new quantum shell is greater than the increase in nuclear charge, so overall there is an increase in electron energy and a decrease in ionisation energies moving down the group.