Covalent bonds are bonds where electrons are shared between atoms to make a full outer shell of electrons on each atom. A good example of a covalent bond is H20 or water. Hydrogen has only 1 electron and as the electrons configuration goes 2,8,8,2, hydrogen needs an electron to fill its outer shell.
Oxygen has 16 electrons, meaning it needs two electrons to fill its outer shell. Therefore, each hydrogen atom shares one electron each and the oxygen atom shares 2 electrons. This makes each atom have a full outer shell.
Covalent bonds usually involve two non metal elements and will involve atoms that do not have a full outer shell.