Each carbon atom in graphite is directly linked to only three carbon atoms through covalent bonds. Therefore, of the four valence electrons in a carbon atom only three are used for bonding and the fourth is relatively free and can move from one carbon atom to other. This results in the conductivity of graphite.
In diamond every carbon atom shares four of its available electrons with other carbon atoms, forming a tetrahedral network. This network is very strong and rigid, thus the diamond is hard. Thus, the bond between carbon atoms in diamond is very strong and hence diamond has a high melting point.