What are fossils?
The word fossil derives from latin. Actually from the word fossus, variously meaning: something dug up, mined or quarried after having been buried. That definition is a pretty good start. More practically in English or paleontological or archeological vernacular, fossils are things dug up that contains the trace evidence of a former living organism, including the organisms itself, or evidence of its existence (even molecules, tracks or poop). All this is notwithstanding the fact that something buried can be exposed by natural forces such as erosion, and, infact, many fossil discoveries are due to natural exposure processes.
Does something have to be a minimal age to be a fossil?
Nope, albeit the processes called fossilization can take a very long time. But, fossilization does not have to be complete to meet the definition of a fossil. Fossilization is all about chemistry. As chemical processes never stop, fossils buried in the earth or under the sea will change ad infinitum, or until hell freezes over. I've read in places on the Internet that something is not a fossil unless it is a certain age, say one million years -- this is bunk. Other sources will say 10,000 years, give or take -- I can live with that.
What is paleontology?
Briefly, paleontology is the study of fossils, and it is both a profession, and a hobby of amateurs like myself. Actually, paleotology entails the study of fossils across geological time, their formation, and especially the study of evolutionary relationships (descendency) between taxa (phylogeny). There is a lot of overlap between paleontology and geology.
How big are fossils?
Fossils can be ginormous to microscopic to molecular. Dinosaurs where many meters long. Entire trees have been fossilized. Ancient bacterial structures have been found is some stromatolites. In some formation molecular traces have been that are consistent with past life forms.
What are index fossils?
Index fossils are fossils are genera or species confined to a certain geologic time range. Thus, if you find it, you know immediately how old is the fossils, and how old is the formation or strata from which it come. Also see: index fossils here.
Are fossils rare?
That's a question difficult to briefly answer. Certainly, the odds of any particular organism becoming a fossil is less than you winning the lottery, far less - see you might be a fossil if - however, a lot of organisms have lived, so many are within the strata. It is reasonable to expect that fossils for most species that have ever lived remain undiscovered, and will remain so forever and ever. One must be careful in that the concept of rare is at best ambiguous, and mostly overused. Still, taking trilobites as an example, among some alleged 20,000 described species, some are only known from parts and pieces. In some formations, a particular species predominates, and others exists in minute proportions. Descriptions such as rarely found, rarely seen, and uncommonly found or seen is often more accurate. Moreover, new species and constantly being found and described; caution is also warranted here, since the concept of species can also be nebulous. For example, many fossil genera are found at multiple fossil sites, in what are now different continents. When they lived, however, they where on the same continents, such as the Supercontinent Panagea. Fossils from different localities are given different names, denoting they come from different places, thought they might not, infact, be different species. Also, they may be rare in one locality (or continent), but not in others.