Steps to address questions
Structure
Bonds
Free electrons
Diamond is very hard and is a poor conductor of electricity as each carbon atom is strongly bonded to 4 other carbon elements by strong covalent bonds in a tetrahedral structure so there are no free electrons to form the "sea of free electrons" and conduct electricity.
However, in graphite, each carbon atom is only bonded by strong covalent bonds to 3 other carbon elements in flat layers of carbon atoms. Hence, there are weak intermolecular forces of attraction "Van Der Waals" forces of attraction between the different layers of carbon atoms so this allows the layers to slide over each thus making it slipper. As each carbon atom is only bonded to 3 other carbons, there is a free delocalised electron in each carbon atom which is free to move and conduct electricity.