The atomic number, otherwise known as proton number, of an element is the number of protons in the nucleus. Scientists represent it with the symbol Z. Atomic number is the traditional name, although personally I prefer "proton number" because it is more obvious what that represents.
All elements have a unique associated atomic number, from the ubiquitous hydrogen with Z = 1 to the synthetic element meitnerium with Z = 109. Such an enormous nucleus is clearly very unstable, and this element has only existed for a fraction of a second, inside a particle accelerator. Because it is purely the atomic number that determines which element an atom is, modern technology means that the alchemist's dream of turning one element into another is theoretically possibly; but the impracticality of blasting protons out of the nucleus with a laser means that there is no commercial value in this possibility.
The elements in the periodic table are ordered by increasing atomic number. Of course, this criterion alone would place them in a one-dimensional array (a straight line), but we form a table by putting elements with similar properties in the same column, and atomic number increases across each row. The value of proton numbers is that we now know that our periodic table has no gaps, although the particle physicists are always creating new elements to add to the end of the table.
Of course, protons are not the only sub-atomic particle in the nucleus. We also have neutrons, which have mass comparable to the proton but no charge. If we arrange the periodic table by atomic number, we get the neat progression we see today. However, if we use the mass number, which measures the number of protons and neutrons (collectively "nucleons") in the nucleus, we get a very confused order indeed. It took quite some advance in science before scientists appreciated the existance of uncharge particles in the nucleus.
As I have mentioned before, the only factor in deciding which element an atom belongs to is the number of protons in the nucleus. Therefore, atoms can have different numbers of neutrons and still be the same element; the neutrons make no difference to the chemical properties, so this is a convenient definition. These varients within each element are called isotopes. For example, the isotopes of hydrogen are known as hydrogen, for the common neutron-free atom, deuterium, which has a single neutron and is used in heavy water, and the even rarer tritium, with two neutrons.
Atomic masses as printed in chemistry data books take into account the extra masses of the isotopes and their relative frequencies. Therefore a typical large sample of the element would have an average atomic mass corresponding to that printed, although no one atom has a mass of 35.5 grams per mole.
As if to illustrate that our periodic table really is being enlongated all the time, both Pferdina and Gritchka have informed me that element 105 is no longer the highest atomic number we have yet achieved. That honour now belongs to Meitnerium, and I have updated the information above. Apparently it is named for physicist Lise Meitner. Thanks to both of you. | 
