Why does the first ionisation energy of nitrogen exceed that of oxygen?

The trend does generally increase across a period, but nitrogen and oxygen show an exception because of electron configuration stability. Nitrogen has the configuration 1s2 2s2 2p3, which is a half-filled 2p subshell, with one electron in each of the three p orbitals. Half-filled subshells are extra stable, so removing an electron from nitrogen needs more energy. Oxygen has 1s2 2s2 2p4, so one orbital now holds a pair of electrons. These paired electrons repel each other, making it easier to remove one. So oxygen loses an electron more easily, giving it a slightly lower first ionisation energy than nitrogen. This same logic explains why beryllium (full 2s) exceeds boron. Watch for these half-filled and fully-filled stability dips in JEE questions.