About 420 BCE, Leucippe of Milet first suggested the idea that matter was made up of "atoms", derived from the greek "a-tomos", meaning indivisible, and that the atoms were in constant motion. His disciple, Democrite of Abdere, took this idea and expanded on it. That they were invisible because of their small size, were solid, were surronded by empty space (to explain movement and density changes), had an infinite number of shapes, and were eternal, because they were perfect. The theory was not accepted by more notable philosophers of the time, including Aristotle, and the theory was ignored, left off to the side. In the meantime, the idea of the four elements - air, fire, earth, and water reigned.
The Greeks were not the only ones with this idea - there was also a philosophy in India, "Vaiseshika", which taught that matter was formed from indestructible particles, and that their assembly into visible, tangible objects were temporary, and that the assemblies would degrade. Eventually, their bonds would dissolve, the particles would need to rest, and then could reassemble into a new world. From the Bhagavad Gita:
"The phenomenal world or material world is also complete in itself because the twenty-four elements of which this material universe is a temporary manifestation are completely adjusted to produce complete resources which are necessary for the maintenance and subsistence of this universe. There is nothing extraneous, nor is there anything needed. This manifestation has its own time fixed by the energy of the supreme whole, and when its time is complete, these temporary manifestations will be annihilated by the complete arrangement of the complete. There is complete facility for the small complete units, namely the living entities, to realize the complete, and all sorts of incompleteness are experienced due incomplete knowledge of the complete."
Eventually, the theory came back into view, this time with experimental evidence to support the idea. In 1808, a book published by John Dalton, "A New System of Chemical Philosophy", revealed experimental conclusions that all atoms of an element had the exact same size and weight, and that all molecules were formed by simple ratios of atoms.
In 1869, russian scientist Dimitri Mendeleïev first constructed a table which classified elements according to their chemical properties. Eventually, this table would become the Periodic Table.
In 1897, Sir Joseph John Thomson made the first discovery that atoms were not indivisible, but also made up of sub-atomic particles, by discovering the electron. This was done using a cathode-ray tube, showing that the cathode ray was actually negatively charged particles smaller than an atom. His first proposed model of the atom was known as "Thomson's Pudding", or more commonly, the "plum pudding model" - as the model suggested that there was a sphere of positively-charged material, and that negatively-charged electrons were mixed into the sphere.
In 1911, Ernest Rutherford discovered the existence of the nucleus of the atom, due to an experiment that found that alpha particles sent into matter would sometimes reflect, and sometimes pass right through. He first explained the internal structure of the atom as a positively charged nucleus, surrounded by negatively charged electrons, which orbited the nucleus like planets orbiting a star, with an attractive electric force representing the role of gravity. The model was also called the "atomic planetary model". (See: Rutherford model of the atom)
In 1913, Niels Bohr expanded on this, as the previous theory hadn't been able to properly handle cases where electrons absorb and emit light. Bohr proposed the theory that electrons could only exist in certain orbital energy states around the atom, and that by jumping between orbits, they could absorb and/or release photons. (See: Bohr model of the atom)
In 1919, Rutherford refined the concept of the nucleus. He discovered the existence of the proton, and that the nucleus contained those protons. In 1932, Sir James Chadwick discovered the existence of the neutron as also existing in the atomic nucleus, which helped a lot with the theories of the atom - as up to that point, scientists couldn't explain the weight differences between different elements. The two particles were known as "nucleons".
The atom is the smallest piece of an element that still contains all the properties of that element. All normal matter is made up of these elements. Currently, about 90 different atoms have been found to exist in nature, and about 118 (at last count) elements have been defined, with the difference being due to ones created in the lab in particle accelerators.
Atoms are very small - the smallest, the hydrogen atom, is about 10-8 cm in size - and most of the atom is empty space, between the nucleus, and the electrons.
Atoms are identified mainly by their "atomic number", or the number of protons in the nucleus. This determines the chemical properties of the element. The number of neutrons in the nucleus may also vary - the number of neutrons determines the isotope. The isotope may alter the stability of the atom, and different isotopes may react differently in nuclear reactions, but they don't affect the chemical properties. For example, there are three known isotopes of Hydrogen. The most common form, simply known as Hydrogen, has no neutrons. Deuterium, or heavy hydrogen, has 1 neutron. Adding a second neutron created Tritium, or ultra heavy hydrogen. Atoms have an "atomic weight", which is approximated by the number of nucleous in the nucleus.
Atoms have a number of electrons "orbiting" the nucleus. (Technically, they are more accurately described as a charged "cloud" around the nucleus, representing the most probable locations for the electron, but that's not truly important for basic understanding.) A neutrally charged atom has exactly the same number of electrons in orbit as it does protons in the nucleus, as the two have the same charge. If their is a difference between the number of protons and electrons, an atom is said to have a charge. If there are fewer electrons, the atom has a positive charge - known as a cation. If there are more electrons, the atom has a negative charge - known as an anion.
Two recent types of microscope, the Microscope Electronique à Transmission (or MET) and the tunnel effect microscope have allowed scientists to actually view individual atoms, making them no longer completely invisible. A special version of the tunnel effect microscope, the atomic force microscope, even allows scientists to deposit and move individual atoms - IBM recently used this to spell out their company name with individual Xenon atoms.
A technology on the horizon, molecular nanotechnology, offers the possibility of actually building machines directly out of individual atoms, instead of carving large pieces of matter to small size.
History of the atom, http://www.neutron.anl.gov/hyper-physics/histatom.html
Atom History, http://perso.club-internet.fr/molaire1/e_histoire.html
Atom Development, http://members.tripod.com/~mrbeens/atom.html