It varies depending on the environment where it fossils and the type of organism:
1-Conservation of organs without modification or with discrete modifications (recrystallization). Is the usual case of hard parts (shells, bones, teeth, calcareous algae ...), and exceptionally soft parts in very particular conditions: ahem. mammoths conserved in icy soils of siberia or in hydrocarbons (ozokerites) north of the Carpathians, insects included in fossil resins (baltic amber).
2-Conservation of organs after their replacement by a different subject preserbando its structure with more or less detail. (It is a process called metasomatism) ahem. silicification of trunks of trees, vertebrate bones or the pyritization of ammonites.
3-Conservation in the form of mold of the organism, internal or external to it. It is often the case of ammonites, shells or conches.
The fossilization process is said to start at 100ºC and pressures above 1Kb (kilobar). Conyeva an exchange of atoms, between the original organism and the matter that surrounds it. They are called fossiligenetic processes. Some of the processes are: carbonation, silicification, phosphatization, pyritization, dissolution.