ionic bond
Imagine a metal atom that is desperate to get rid of a loose outer electron, and a non-metal atom that is hungry to grab one. The metal hands its electron over completely. Now the metal is a positively charged ion (a cation) and the non-metal is a negatively charged ion (an anion), and the two opposite charges pull on each other. That electrostatic attraction is the ionic bond, and table salt, NaCl, is the classic example.
More precisely, an ionic bond is the limiting case of bonding in which one atom transfers electrons to another, producing ions whose attraction is described well by Coulomb's law: opposite charges attract with a force that falls off with distance. In a real salt the ions do not pair up two-by-two; instead millions of cations and anions stack into a regular three-dimensional lattice, so that each Na+ is surrounded by six Cl- and vice versa. The bond energy comes from summing all these attractions and repulsions over the whole crystal.
Ionic bonding explains why salts are typically hard, brittle, high-melting solids that conduct electricity only when molten or dissolved (the ions must be free to move). It is one of three idealised bond types — ionic, covalent and metallic — and most real bonds sit somewhere in between. Pure ionic bonding is itself an idealisation: even in NaCl there is a little sharing of electrons, and as the cation gets smaller and more highly charged the bond gains covalent character, which is exactly what Fajans' rules describe.
Sodium (electron configuration ...3s1) gives up its single 3s electron to chlorine (...3p5), forming Na+ and Cl-. The resulting ions both reach a stable noble-gas configuration, and the crystal NaCl holds together purely by the attraction of these charges.
Electron transfer from a reactive metal to a reactive non-metal: the textbook picture of ionic bonding.
There is no such thing as a perfectly ionic bond; even the most ionic compounds have a small share of covalent character, and a bond is best thought of as a point on a continuum, not a category.