An atom is a particle of matter indivisible by chemical means  which form the building blocks of molecules. Although the word "atom" comes from the Greek term for indivisible, átomos, atoms are actually made up of three different kinds of subatomic particles; some of these are composed of yet smaller particles.
In the atomic nucleus there are positively charged protons and electrically neutral neutrons. Surrounding the nucleus are negatively charged electrons. The simplest atom is that of Hydrogen. It has only a single proton and electron and no neutrons.
Each proton and neutron are comprised of three quarks and are contained closely together in the center of an atom, forming the nucleus. The nucleus is extremely dense, typically having a density of 1017 kg m-3, or 1013 that of lead. Electrons move in the space around the nucleus, and are arranged around it in a series of layers, known as shells or energy levels. Since protons and neutrons are approximately 2000 times as heavy as electrons, the vast majority of an atom's mass is found in the nucleus. Currently quarks and electrons are considered truly elementary particles. Atoms are mostly empty space, as the relative size of the nucleus compared to the area of the lowest electron shell is about that of a pea in a stadium. Another common analogy for the atom along the same lines is the "fly in the cathedral", where the cathedral is the whole atom and the fly is the nucleus.
Those who understand electrical theory might notice that positively charged particles packed closely together would repel one another. As the protons and neutrons in the nucleus are hadrons, they are affected by another of the four fundamental forces, known as the strong nuclear force. Unsurprisingly given its name, the strong nuclear force is stronger than the electrical repulsion of the protons at these small scales and so the nucleus stays together. The quantization of this force is a particle called a "gluon".
Chemical elements are made up of atoms with certain properties. The number of protons in the nucleus of an atom (known as the atomic number) determine the properties of the atom, and the element it constitutes. For example, Hydrogen has one proton, and therefore an atomic number of 1. Oxygen has 8 protons in its nucleus and has an atomic number of 8. Under normal conditions, atoms contain an equal number of protons and electrons.
Atoms are normally electrically neutral; they have no charge. However, electrons can be gained or lost (depending on the element and the conditions) to form an ion. An atom that loses electrons becomes positively charged and is known as a cation. An atom that gains electrons electrons becomes negatively charged and is known as an anion. Anions of the Halogens (Group 7 elements) fluorine, chlorine, bromine and iodine are known as fluoride, chloride, bromide and iodide (replacing -ine with -ide), respectively.
Many common substances are made up of ions. For example, sodium chloride (NaCl), otherwise known as table salt, is made up of sodium cations (Na+) and chlorine anions (chloride, Cl-) in equal proportions. The negatively charged chloride ions are attracted to the positively charged sodium ions, forming an ionic bond. This results in a lattice structure, which is responsible for sodium chloride being crystalline in its solid state.
Atoms of the same element (having the same number of protons) that have different numbers of neutrons are known as isotopes. Some isotopes are more stable than others, and occur more often in nature, but there is no "standard" number of neutrons in a given element. The atomic weight of an element is a weighted average of the atomic weights of all naturally occurring isotopes (isotopes that are not radioactive). The atomic weight of an isotope is approximately equal to the number of neutrons and protons (number of protons remains constant in a given element), For example, chlorine has two naturally occurring isotopes: chlorine-35 and chlorine-37, with relative abundances of 75.78% and 24.22%. The atomic weights of those isotopes are 34.96885268 and 36.96590259, respectively. The atomic weight of chlorine is the weighted average, so it is (0.7578×34.96885268)+(0.2422×36.96590259)=35.45293758. So the atomic weight of an element is not necessarily an integer for two reasons: the element as it occurs in nature may be (as in this case) a mixture of two or more isotopes, and the individual isotopes may have non-integral atomic weights due to the mass defect. Mass defects are calculated relative to the carbon-12 isotope, which is defined to have an atomic weight of exactly 12. Other isotopes generally have non-integral atomic weights because the nuclear binding energy ties up some of the mass according to the formula E=mc². See Quantitative Analysis of Alpha Decay. Many isotopes are radioactive and decay over time.
History of the Atomic Model
The early Greek philosophers Democritus and Leucippus first proposed that all matter in the universe consisted of particles, which are the smallest units of matter, though Aristotle used his fame to convince the scientific community that matter consisted of various ratios of earth, fire, water, and air, which can exist in any quantity. This mislead them for centuries.
John Dalton developed a realistic atomic theory based on four principles: that all matter is composed of atoms, that each element's atoms are of one size, atoms combine chemically to form compounds, and chemical reactions occur when atoms are joined or separated. His theory relied on Lavoisier's Law of Conservation of Mass and Joseph Proust's Law of Definite Proportions.
J. J. Thompson later developed the Plum Pudding Model, a result of his discovering the electron using cathode ray tube. Ernest Rutherford later discovered in his gold foil experiment that the atom was mostly empty space, with a nucleus of protons in the center. James Chadwick discovered the neutron and complexities of the nucleus, paving the way for nuclear fission.