The periodic table is a bit oddly shaped but it does make sense in terms of electronic configuration. The horizontal rows are called periods and there are seven of them. The vertical rows are called groups and there are 18 of them.
Elements contained in the same group tend to have similar properties and these different groups have names. Group 1 elements are called Earth metals and readily give up an electron to become plus 1 cations. Group 2 elements are called alkaline Earth metals and readily become plus 2 cations. On the other end of the table we have group 17 which are called the halogens. They readily gain an electron to become minus 1 anions. Group 18 are called the noble gases and are highly stable with extremely low reactivity.
The periods are arranged to make sense in terms of electron configurations. Simply put, electrons are found in 4 basic types of orbitals: s, p, d, and f. These orbitals inhabit different shells, or energy levels, about the nucleus. Period 1 elements have only 1s electrons, hence there are only two. Period 2 elements have the second shell electrons in the s and p orbitals and so on down the line. The elements between groups 3 and 12 are called transition metals and they contain the d shell electrons. Finally, the two outlying rows on the very bottom (often labelled the lanthanide and actinide series) contain the f orbitals.
There are various trends when one goes across a period (from left to right) or when one goes down a group (from top to bottom). Atomic radius (size) decreases across a period and increases down a group. Another trend is electronegativity which increases across a period and decreases down a group. This makes fluorine (upper right corner) the most electronegative element on the periodic table.