The modern use of the word fossil refers to physical evidence of former life from a period of time prior to recorded human history. This evidence can include fossilized actual remains (bones or seeds) of organisms, impressions or molds of their physical form (dinosaur footprints), and marks or traces created in the sediment by their activities.
Archeologists and geologists are professionals who find and study these fossils. They can tell the age of the fossils through a variety of radiometric dating techniques. The breakdown of certain elements such as carbon, uranium and potassium takes place at a known rate, so the age of a rock or mineral containing these isotopes can be calculated.
Fossils can be classified into 10 main groups. The fossil groups are index, trace, transitional, microfossils, resin, derived, wood, subfossil, and chemical fossils.
There are several different ways fossils may form. The two most common ways are permineralization and casts and molds.
Permineralization is a process that occurs when an organism is buried. The spaces that are empty inside the organism are filled with mineral rich water from the ground. Some of these fossils consist only of skeletal remains and teeth while others contain traces of skin, feathers, and even soft tissues.
Casts and molds happen when the original remains of the organism completely dissolve or are destroyed in another way. The mold or organism shaped hole left in the rock is called an external bold. If this hole is later filled with minerals, it then is classified as a cast.
Other ways fossils can be formed are: Authigenic mineralization, replacement and recrystallization, adpression, soft tissue, cell & molecular preservation, carbonization, and biommuration.
Sources : BBC Nature, DiscoveringFossils.Co.Uk, Wikipedia