An atom is the smallest possible physical unit of a chemical element. Atoms consist of protons (with a single positive charge each), electrons (with a single negative charge) and neutrons (with no charge). For example, the most common isotope of hydrogen has zero neutrons and consists in its atomic form of a single proton and a single electron. All heavier nuclei require neutrons to stabilize them.
The atoms which comprise each individual element have of a unique number of protons, known as the "atomic number". Most elements occur in nature as several different "isotopes", each with a different number of neutrons. Combining the number of protons and the number of neutrons results in the atom's "mass number". For example, chlorine is found primarily as two stable isotopes, with mass numbers of 35 and 37, but both have 17 protons.
Often, all but one (if not all) of these isotopes will be unstable, and will decay at a predictable rate to one of the lighter, more stable isotopes. Isotopes are referred to by a number equal to the total number of protons and neutrons (i.e., U-235, U-239, etc.) which is also very close to the actual atomic mass. Elements with no stable isotopes eventually decay into other, lighter elements. The relative stability — that is, how long it takes a given percentage to decay — and initial ratios of the isotopes will determine their ratios at a subsequent point in time. This is the basis for radiometric dating.
The neutrons and protons form the nucleus, a very dense core which contains virtually all the mass of the atom. Each atom also technically comes with a number of electrons equal to the number of protons, making it electrically neutral. However, in nature, the electrons can be more or less closely associated with the nucleus and its protons.
While the atom is normally thought of as being like a little solar system, with the nucleus represented by the sun and the planets representing the electrons in "orbit" around it, it is really more like a tiny positively charged glob with a series of uniquely configured negatively charged force fields pulsating around it.
In the state of matter called plasma, nuclei and their electrons are completely dissociated. This only occurs at very high temperatures.