To put it simply, covalent bonds are bonds in which each atom contributes one of it's unpaired valence electrons. This can be seen by drawing out the Lewis structure.
Ionic bonds are those in which atoms donate or receive electrons and the resulting anions and cations are attracted through electrostatic attraction. For example, neutral sodium will donate one of its valence electrons to chlorine to give the halogen a stable octet. Na has lost an electron and is now a sodium cation (Na+), while the chlorine gains an electron and is now a chloride anion (Cl-).