Diamond tip tools are used in a fashion that grinds like sandpaper. The diamonds are small bits and dust that is on the surface of the substrate (cutting tool). The diamonds are not machined to a specific shape, but are selected by size. There are a few technologies for keeping the diamonds on the surface, but the basic idea is glue. Different substrates and glues are used depending upon what you are cutting.
Another technology for tools is carbide tips. Carbides are salt-like compounds formed from carbon and a metal. Carbides comes in many forms, but the most common for cutting tools is tungsten carbide and titanium carbide. Carbide is almost as hard as diamond, but much cheaper since it can be manufactured instead of mined. Again, the difficulty of using it on cutting tools is how to attach chunks of carbide to the business end of the cutting tool. Carbide can be shaped to fit and brazed in place. See link with picture. You can see the tip of a saw blade shaped and sharpened. The blade tip is bonded to the wheel with brazing.
The oldest technology is high carbon steel. In this case, the entire tool is made of steel with between 0.5% and 1% carbon content. The steel can be made soft through heat treatment process called annealing. The steel is heated to a high temperature and then cooled very slowly over many hours. Once annealed, the steel can be machined and worked to the shaped desired. Afterward, the steel is put through another heat treatment to re-harden by heating the steel and then cooling very quickly by quenching in oil or air. After the hardening, the cutting edge is sharpened and honed to the desired cutting edge.