In what order are the elements of the periodic table arranged?
In numerical order (left to right), they are arranged by the number of protons in the nucleus of a single atom of each element.
Aside from that, you'll notice that the rows have different numbers of elements in them. The first row only has two: hydrogen and helium. That's because 2 is the number of electrons that can fit in the first orbital shell around the nucleus. Elements whose atoms have naturally full outer shells are called inert.
The 2nd orbital can hold 8 electrons, and so there are 8 elements in the next row of the periodic table. But wait, why are there 2 elements on the left of the row, and 6 on the right?
This is done because the columns of the table are grouped as well; elements in a single column have similar behaviours. Either they interact with other elements in similar ways, or they have similar properties.
For example, the first column (starting at lithium-3) is called the Alkali metals, and the last column is called the noble gases. The second-last column (starting at fluorine-7) is called the halogens. Halogens mix with metals to produce different kinds of salts, like how chlorine-17 mixes with sodium-11 to produce NaCl -- ordinary table salt.
The periodic table is chock full of patterns and interesting things like this. Check your chemistry textbook for more cool stuff about it.