An ion is an atom with a net positive or negative charge. Ions occur when the number of electrons around a nucleus doesn't exactly equal the number of protons in the nucleus. The number of neutrons in the nucleus has no effect on whether or not a given atom is an ion since they have no charge, although changing the number of neutrons does change the isotope.
Ions are very chemically reactive (that means that they easily undergo chemical reactions), which can cause problems in biological systems. There are a number of chemical ways to produce ions, like dissolving salt in water, or the process of combustion. Additionally, radiation can produce ions if the radiation has enough energy; this is referred to as ionizing radiation.
To learn more about ions please see UC Davis's chem wiki.
Phet: Build an Atom
Below is a interactive PhET simulation from the University of Colorado. This simulation creates an atom from protons, neutrons and electrons and test skills with the periodic table.