Key Concepts:
period   nonmetal   Group or family   valence electron    metal   atomic mass
In 1869 when Dmitri Mendeleev developed the first really useful periodic table, he organized it using the atomic masses of elements. He used masses because the Proton would not be discovered by Ernest Rutherford, in his famous Gold Foil experiment, for another 40 years. The present day periodic table uses an arraignment that is based on the atomic numbers of the elements. As of 2006, the periodic table contains 117 confirmed elements. Ninety of these elements are found naturally on Earth. The rest are synthetic elements that have been produced artificially using particle accelerators. Though elements 43 (technetium) and 61 (promethium), have lower atomic numbers than the naturally occurring element Uranium, with atomic #92, they are synthetic. Elements 93 (neptunium) and 94 (plutonium) are listed with the synthetic elements, but have been found in trace amounts on earth.
 
Watch this video for more information on the arrangement of elements in the periodic table.
The periodic table, is just that, a table of elements that are arranged to show periodic trends ( A trend is a pattern of change.) in the structure and properties of elements. On the periodic table, vertical columns are called groups or families, while horizontal rows are referred to as periods. Groups are numbered left to right, 1 through 18. They can are also named for the top element. For instance, group 16 is also called the Oxygen group, because Oxygen is at the top of the group.
Moving from left to right within a period, the atomic number and the number increases by one for each subsequent atom. The  number of valence electrons also increase by one. ( The transition Metals in groups 3 through  12 are an exception. You'll need to know more about electron energy levels to understand why.) As the number of valence electrons increases, the atomic radius ( size) of the atom increases also. Generally, each element within the same group has the same number of valence electrons. This is one reason why elements within the same group have very similar chemical properties. ( This means they pretty much act the same way in chemical reactions.
Moving down within a group, increasing period numbers correspond to filled valence energy levels. This means that in period 1 the first energy level is being filled with electrons. In period 2 the second level is being filled. In period 3 the third energy level is filled. This trend continues for each successive period. All atoms within the same period have the same number of energy levels and the level fills with an increasing number of valence electrons from left to right through the group.

Figure 3
Figure 3 above illustrates these trends within the periodic table. First, note that the Transition Metals are left out of this diagram. The diagram shows the first 2 Periods of the Periodic table. The Period Numbers are shown at the left, in black. Groups 1, 2,      13,14,15, 16, 17, and 18, are shown above the diagram in red. Only the valence electrons are shown in black. Inner electrons are in green.
    All atoms in group 1 have 1 valence electron. All atoms in group 2 have 2 valence electrons. All atoms in group 13 have 3 valence electrons and so on, all the way to group 17 with 7 valence electrons. In group 18, with the exception of Helium, all atoms have a filled valence level with 8 valence electrons. Helium has only 2 valence electrons because it is at the end of the first period and 2 valence electrons fills the first energy level.  Beginning with the second period, the second energy level starts to fill. By the end of the second period the second energy level is filled with a full compliment of 8 valence electrons...a stable octet.
 
The periodic table can be downloaded from the following website:
http://ed.sc.gov/agency/offices/assessment/Programs/endofcourse/periodtable022304.pdf