For coal, the source material is primarily land based plants, preferentially those that were growing in swamps or other similar environments. There is an oft repeated claim that coal only formed during the Carboniferous because the lack of decomposers able to digest lignin, but this is demonstrably false. As described in this FAQ, the abundance of coal formation during the Carboniferous relates to specific paleoclimatic and paleogeographic conditions (i.e., lignin decomposers were present at this time) that supported large areas of swamps for significant periods of time. For petroleum and natural gas, the source material is primarily marine photosynthetic microorganisms, like algae, phytoplankton, etc.

For both types, in order for the respective fossil fuel to form, the raw organic (in the chemical sense) material needs to be buried quickly and experience the right temperature and pressure conditions for the various reactions that produce either coal or petroleum. The wiki articles on coal and petroleum formation are decent and provide a decent overview of the source material and the relevant chemical processes along with the specific conditions required for them to occur.