This question is designed to test a students knowledge of electronic configuration in elements, and their ability to use this knowledge to assign a reasoning to the trend given. The general trend is an increase of first ionisation energy along the period. This can be explained by considering the increase in nuclear charge as protons (protons are positive) are added to the nucleus and incomplete shielding. Shielding is a term that describes the repulsions between electrons that arise because they have the same charge (electrons are negative), and like charges repels one another. As the nucleus becomes more highly charged there is a greater attraction between the nucleus and the electrons, as opposite charges attract. The increase of attraction also causes the atom to contract. These to factors, the increases in electrostatic attraction and the reduced atomic radius, mean more energy is required to remove the electron and so the first ionisation energy increases.
There are, however, some data points that seem anomalous. These are the decreases from Be to B and the decrease from N to O. This is where a more intimate knowledge of electronic configuration is needed. Be is 1s2 2s2 and B is 1s2 2s2 2p1. The outermost electron is now in a p orbital. Electrons in this orbital are, on average, further from the nucleus. (I would take this as an opportunity to talk about the respective shapes or the orbitals, ideally drawing them) This means that less energy is required to remove the outermost electron, and so there is a decrease in the ionisation energy. The second decrease (N to O) is due to the p orbital now having two electrons in it. (I would take this as an opportunity to discuss the Aufbau principle, Pauli exclusion principle and Hund's rule of maximum multiplicity, as these concepts need to be understood for this question and many others in inorganic chemistry.) As the p orbital in O has two electrons in it there is more electron-electron repulsion and this destabilises the orbital, meaning less energy is needed to remove it.