Covalent compounds contain atoms that share electrons - for example, hydrogen chloride (HCl) share electrons (one from each atom), in order to achieve a full set of 8 electrons in their outer shell. Covalent compounds can either be simple, such as HCl, or form giant lattice structures - such as diamond and graphite.
Ionic compounds, on the other hand, contain atoms that do not share electrons - one atom gives an electron (or a few) to another atom. NaCl, for example - your regular table salt - is an ionic compound. The sodium atom, which has only 1 electron in its outer shell, donates its electron to a chlorine atom, which contains 7 electrons in its outer shell.
When the chlorine atom receives the sodium's electron, its outer electron shell becomes full (with 8 electrons). The sodium atom now also has a full outer shell now. Both are no longer atoms and form ions as they have an unequal number or protons and electrons. They are attracted to each other by electrostatic attraction - the positive charge of the sodium atom attracts the negative charge of the chlorine atom.
Ionic compounds form giant lattice structures and are good conductors of electricity, as the ions are free to carry a charge.