What Ionic Bonding Is
Ionic bonding is the type of chemical bonding that binds metals with non-metals*, forming ionic compounds. An ion is just an atom (or sometimes a molecule) with an overall electric charge - many atoms and molecules have exactly as many electrons as they have protons, so the charges cancel out; when that doesn't hold true, we end up with ions.
Metals are prone to losing electrons from their outside shell, leaving them with a positive charge; non-metals often pick up additional electrons from somewhere, filling up their outside shell and leaving them with a negative charge. Opposite charges attract, so electric forces tend to cause these positive and negative ions to stick together. Since those forces radiate out in all directions, you don't just get one positive ion (or cation) bonding with one negative ion (or anion) - any more ions that happen by get pulled in, too. There's always a sweet spot where the pushing and pulling of the ions balances out, allowing new ions to slot neatly into any existing structure. That neatness gives a very regular lattice-like pattern to the solid - in other words, ionic compounds form crystals.
What Ionic Bonding Isn't
It's worth saying something about some common misconceptions about ionic bonding. If you have learned about it before, you may have been told that an ionic bond is what you get when a metal ion donates an electron to a non-metal. This description has a pleasing simplicity to it, but it is really very misleading. For one thing, ionic bonding typically holds together many atoms at once. This is in contrast to the covalent bonds** that hold non-metals together, where the bonding is down to each atom sharing electrons with its neighbours, which leads to the formation of well-defined molecules. Ionic compounds are not really made of molecules at all, just big crystalline structures.
The other thing wrong with the electron-donation picture is that the ions have usually gained or lost electrons long before they ever meet - for many elements, like sodium and the other alkali metals, it is rare to find them any other way on Earth. Less reactive metals may have been exposed to ionising radiation, or lost an electron or two in a collision. Reactive non-metals have a tendency to pick up any free electrons they bump into, whatever the source, because they fit nicely into the geometry of their outside shells.
Ionic compounds are characteristically hard, usually with high melting points, and very brittle. The hardness and high melting points are down to their crystal structure; as long as the lattice holds, they are solid and quite strongly bonded. However, since the crystal is made of alternating positive and negative ions, a knock that causes one layer to get out of alignment with the next will often lead to cations lining up with cations, and anions with anions, producing a repulsive force that tears the crystal apart - hence the brittleness. Metals, which also have a crystalline structure, don't suffer from this problem, which is why they are much more malleable.
Many ionic compounds are soluble in water. This is because water molecules are polar, in the sense that they have more positive charge on one side than the other. A negative ion will attract the positive ends of water molecules, and when it collects enough water molecules that way, their collective attraction can overcome its bonding with its ionic neighbours and carry the ion away. The positive ions dissolve much the same way. All these positive and negative ions allow a solution to conduct electricity - distilled water is actually an electrical insulator, whereas salt water conducts extremely well. There is a useful complication to the way ion solutions conduct electricity - because the charge is carried by two kinds of ions travelling through space, not just free-floating electrons like you get in a metal, they tend to separate over time - cations are attracted to cathodes, and anions to anodes. This process, known as electrolysis, makes it possible to extract the constituent elements of a salt; sodium, potassium, calcium and various other elements were first isolated in this way.
*Strictly speaking, some ionic compounds are formed entirely from non-metals, with polyatomic cations taking the place of the metals, but usually you can assume that ionic bonding joins metals with non-metals.
**We should note here that there is not really a sharp distinction between covalent and ionic bonds. Many covalent bonds are polar, meaning that the electrons are shared unevenly between the atoms, so that one of the atoms acquires a positive charge, and the other a negative one - these bonds can be considered to be a bit ionic. Similarly, ionic bonds can be considered mildly covalent when electrons get shared between atoms, which they inevitably do. Metallic bonding is sometimes considered a form of covalent bonding, but sometimes not - the shared electrons are more like a sea than a set of pairs. Chemistry gets pretty messy when you look close enough.
This writeup also appears on my blog with an 'understand-o-gram' by Sonya Hallett.