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. Spinning around the nucleus are negatively charged electrons. Hydrogen, in its most common isotope, has only one proton and no neutrons.
Protons and neutrons are comprised of quarks and are contained closely together in the center of an atom, forming the nucleus. 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. The nucleus stays together because of what is known as the strong nuclear force. The quantization of this force is a tiny 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 in the valence shell 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 common 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 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 number of neutrons and protons (number of protons remains constant in a given element) in all naturally occurring isotopes. Many isotopes are radioactive and decay over time.