All materials are made up of atoms arranged in many different ways. Conductors and insulators share this ultimate likeness at the atomic level. For example, wood, an insulator, is made up of carbon, hydrogen and oxygen atoms arranged in a particular structure to give the material called wood. A material like niobium oxide, a conductor, contains atoms of niobium and oxygen. Here the structure is different, but the basic building block in conductors and insulators are atoms.
Hardness and softness are features shared by conductors and insulators. For example, sulfur is an insulator and is soft. Sodium, a metal, is a conductor and is also soft. On the hard side, we have Iron, which is a conductor, and glass, which is a hard insulator.
Density is a measure of how heavy a material is or how closely packed the atoms are. High-density materials can exist as conductors or insulators. For example, lead, a conductor, is a high-density material. So also is lead oxide, an insulator.
Appropriate doping of an insulator can make it a semiconductor or even a superconductor. An example is lanthanum copper oxide, a ceramic insulator. In 1986, George Bednorz and Alex Muller doped it with a little Barium and it became a superconductor with a high transition temperature. They got the Nobel Prize in Physics in 1987 for that chemical trick of converting an insulator to a superconductor via doping. Similarly, a conductor can be made to become an insulator through doping. Aluminum is a conductor. Doping aluminum with oxygen gives aluminum oxide, an insulator.