These are studied in Organic Chemistry and Biochemistry. As opposed to ionic bonds, which are due to coulombic forces between opposite charges-not directional, covalent bonds are localized between two atoms, and have a definite shape. The covalent bonds exist between the yellow balls. The theory to explain bonding in metals is called The Band Theory. It is based on quantum mechanics.
Metals usually have a filled shell plus one or two extra electrons.
Extra electrons in metals are not restricted to one atom but can move throughout entire structure in what is called an electron sea. Metal atoms are positive charges in the electron sea. They are held together by the electrons. The electrons are free to move. Thus metals are good conductors of heat and electricity.