Trends in the periodic table

The periodic table is named because trends were observed periodically among the elements, which are organized accordingly. Groups (columns) on the table are defined by the number of valence electrons in an atom. Periods (rows) are defined by the number of complete shells the atom has.

Ionization energy
Ionization energy refers to the amount of energy required to remove one electron from an atom. Intuitively, it increases from left to right. The alkali metals (group 1) have low ionization energies because it would leave them with a full, stable shell. Conversely, noble gases have high ionization energies because they already have a stable shell.

Ionization energy also decreases going down the table because electrons farther from the nucleus are less bound by the attractive force between protons and electrons.

Atomic radius
Atomic radius refers to half the width of an atom. It decreases from left to right. Since atoms in the same period contain the same number of shells, and vary only with the number of electrons in the outermost shell, the width of the atom is unaffected. It is only through the addition of shells that the atom becomes wider. Thus, ionization also increases going down the table.

Electron affinity
Electron affinity refers to the energy difference in a gaseous atom after receiving an electron. This means the number will be negative or close to it and represents the relative energy of the atom with one extra electron to that same atom without the electron. Atoms, like everything in nature, tend towards the lowest energy state possible. Noble gases have a low electron affinity while halogens have the "highest" electron affinity. The actual value of electron affinity for halogens will be in the negatives since they lose so much energy after filling their valence shell. Electron affinity increases from left to right, dropping to 0 at the noble gases, and decreases going down the table.

Electronegativity
Electronegativity refers an atom's attraction for bonding electrons in a chemical bond. Atoms with high electronegativities will attract orbiting electrons more in a bond, creating a polar bond. Electronegativity increases from left to right, and decreases going down the table. The attraction of an atom to electrons is dependent on the number of valence electrons (group number) and the number of shells (period number). Atoms with valence shells close to being completed will attract electrons more.

Atoms with more shells will not exert as much of an attraction on electrons because of electron shielding, a process where shells closer to the nucleus diminish the attraction of nucleus. Also, atoms with more shells will have greater atomic radii, meaning electrons are farther from the nucleus anyway, and thus be attracted less.