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The nucleus of an atom is consisted of neutrons and protons (actually a lot more complicated then that but this will work for now).
One field of chemistry is the study of properties of matter. The chemical properties of any element is dictated by its electronic configuration.
Number of electrons an atom greatly determines how its electronic configuration will be and this dictates what chemical reaction will be with other atoms.
If 2 different atoms having the same exact chemical properties, we cannot distinguish between chemically. However, one can be "heavier" than the other due to some extra weight (extra neutrons). These 2 atoms are thus, "the same" chemically, yet "different" physically.
The periodic table is arranged in chemical means. So if 2 atoms having the same chemical properties, they are the same element.
These are called isotopes. The word means "iso": the same; "tope": place. The word by itself in Greek means "the same place". They are chemically equal but there are different.
There are rules governing how much number of electrons, protons and neutrons an atom can have. For a stable atom, the number of electrons has to be equal to the number of protons. However, the number of neutrons are somewhat varies. It can have 0, 1, 2, 3 or more (there is a limit due to EM, weak and strong forces).
So, H can have 0, 1 or 2 neutrons but they are all chemically the same and thus the same element. However, they are different because some of them have more neutrons. Usually, scientists will put the mass number next to the element's symbol such as U235 or U238 to differentiate them. Hydrogen is special and H0 is just hydrogen, H1 is deuterium and H2 is tritium (A-bomb related).
For more info:
http://en.wikipedia.org/wiki/Isotopes_of_hydrogen
http://en.wikipedia.org/wiki/Isotope