Why is the periodic table so uniquely shaped?

Dear scientist,
     I was wondering if you could help me to understand why the periodic 
table has such a unique shape, and what is that suppose to tell us about 
atoms? Any information you could pass along would be appreciated, thanks.
                                             Sincerely,
                                                   Carrie

The periodic table need not have any particular shape as long as it properly represents what it is supposed to represent: the chemical properties of the elements.

Elements on the periodic table were initially grouped according to the ratio of the element to the oxygen in the element's oxide. For example, elements such as lithium and potassium, which form oxides according to E₂O, were in one group; elements which form oxides according to EO (such as copper, zinc, calcium) were in another group; elements which form oxides E₂O₃ (boron, aluminum, iron) in yet another; and so forth.

Before long we realized that not everything is the same. For example, iron forms two different oxides: FeO and Fe₂O₃; while carbon only forms one oxide, CO₂. But subtle distinctions aside, the elements were grouped according to their chemical properties -- and they still are. Now the table has a slightly different organizing principle (electronic structure and atomic number) but since chemical properties follow this closely, the table is still grouped according to chemical properties.

The essential design elements (though it wasn't humans who designed it! they are dictated by chemical and physical properties!) in the periodic table are as follows:

1. Elements in any particular column must have the same number of valence electrons, that is, electrons in the outer subshells.
2. The first row has only two elements, because only two valence electrons are allowed in the first shell. The elements in this row don't fit perfectly into any of the columns of the table (though helium is a better fit to Column 18 than hydrogen is to Column 1).
3. The second row has only eight elements, because only eight valence electrons are allowed in the second shell.
4. The third row has only eight elements, because, while the third shell holds 18 electrons, the last ten get mixed up with the first two electrons of the fourth shell.
5. The next two rows have eighteen elements each, of which the first two and last six fit under the eight columns in the second and third rows. This means that the second and third rows have elements in columns 1, 2, 13, 14, 15, 16, 17 and 18 but not in between. These used to be called IA, IIA, IIIA, IVA, VA, VIA, VIIA and VIIIA. Columns 3-12 ("transition elements") used to be called IB, IIB... and fit into the transition between columns 1-2 and 13-18.
6. Rows six and seven have an additional 14 elements ("rare earths" or "second transition elements") each, which fit between Columns 3 and 4.

But as long as the table preserves the design constraints listed above (which keep elements in groupings by chemical properties), it can have any sort of shape. Some alternative versions (not all of them holding to the design elements chemists think are important) are found here.

References can be found here. You should also check out WebElements.