Cations and anions do not actually represent a periodic trend in terms of atomic radius, but they do affect atomic radius, and so we will discuss them here.
A cation is positively charged, meaning that it is an atom that has lost an electron or electrons. The positive charge of the nucleus is thus distributed over a smaller number of electrons and electron-electron repulsion is decreased, meaning that the electrons are held more tightly and the atomic radius is smaller than in the normal neutral atom. Anions, conversely, are negatively charged ions: atoms that have gained electrons. In anions, electron-electron repulsion increases and the positive charge of the nucleus is distributed over a large number of electrons. Anions have a greater atomic radius than the neutral atom from which they derive.
The process of gaining or losing an electron requires energy. There are two common ways to measure this energy change: ionization energy and electron affinity.
The ionization energy is the energy it takes to fully remove an electron from the atom. When several electrons are removed from an atom, the energy that it takes to remove the first electron is called the first ionization energy, the energy it takes to remove the second electron is the second ionization energy, and so on. In general, the second ionization energy is greater than first ionization energy. This is because the first electron removed feels the effect of shielding by the second electron and is therefore less strongly attracted to the nucleus. If a particular ionization energy follows a previous electron loss that emptied a subshell, the next ionization energy will take a rather large leap, rather than follow its normal gently increasing trend. This fact helps to show that just as electrons are more stable when they have a full valence shell, they are also relatively more stable when they at least have a full subshell.
Ionization Energy Across a Period
Ionization energy predictably increases moving across the periodic table from left to right. Just as we described in the case of atomic size, moving from left to right, the number of protons increases. The electrons also increase in number, but without adding new shells or shielding. From left to right, the electrons therefore become more tightly held meaning it takes more energy to pry them loose. This fact gives a physical basis to the octet rule, which states that elements with few valence electrons (those on the left of the periodic table) readily give those electrons up in order to attain a full octet within their inner shells, while those with many valence electrons tend to gain electrons. The electrons on the left tend to lose electrons since their ionization energy is so low (it takes such little energy to remove an electron) while those on the right tend to gain electrons since their nucleus has a powerful positive force and their ionization energy is high. Note that ionization energy does show a sensitivity to the filling of subshells; in moving from group 12 to group 13 for example, after the d shell has been filled, ionization energy actually drops. In general, though, the trend is of increasing ionziation energy from left to right.