Synthetic fuels are any liquid fuel manufactured from something that has energy - like coal, natural gas, tar sands, or biomass. By contrast, a natural fuel would be something like oil. Synthetic fuels are frequently referred to as synfuels.
In 1944, when the Synthetic Liquid Fuels Program was established as a program run by the US Bureau of Mines, the United States government supported the production of synthetic fuels with investment. As a result, there has been a major impetus for gas and oil companies to use them, as many do - for example Shell, Exxon, Statoil, Rentech, and Syntroleum. However, this synthetic fuels program was halted by Congress in 1985, after spending $8 billion US Dollars (USD) over a 40 year period. In the 1970s, a synfuel tax credit program was established, continuing the push towards synthetic fuels. TECO, Progress Energy, DTE, and Marriott have taken advantage.
Synthetic fuels are classified based on what feedstock was used to create them. By far, the three most prominent processes are Coal-To-Liquids (CTL), Gas-To-Liquids (GTL) and Biomass-To-Liquids (BTL).
The widest-used form of synthetic fuel is liquefied coal and its derivatives. The Fischer-Tropsch process, developed by Nazi Germany and used by Sasol in South Africa today is one of the most used for converting coal, as well as biomass or natural gas, into synthetic fuels.
Whatever you're using to produce the synthetic fuels, the first step is to convert it into carbon monoxide and hydrogen gas. In the gas of natural gases like methane, this requires partial combustion. For coal or biomass, gasification is necessary, which means combining the material with water and oxygen or air at high temperatures. The carbon in the coal combines with oxygen in the water to create carbon monoxide, while the leftover hydrogen atoms create hydrogen gas. This produces a substance called syngas, which can itself by used as a fuel, or further processed into diesel or another derivative.
Diesel used to be made only by petroleum, hence it was called petrodiesel. Today, other substances such as biomass or natural gas are used to make it, hence biodiesel.