Carbon can form several different structures, and they have very different properties. In diamond, each carbon atom forms four covalent bonds with four other carbon atoms. The atoms are arranged in a regular, three-dimensional pattern that carries on in every direction. This is called a giant covalent structure.
Covalent bonds are strong, and every atom in the crystal is held in place by four of them. A great deal of energy is needed to break so many strong bonds. This is why diamond has a very high melting point. It is also why diamond is very hard, which makes it useful for the tips of cutting tools and drills.
All four outer electrons of each carbon atom are used in bonding. There are no free electrons and no ions that can move, so diamond does not conduct electricity.