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The fossil record is the list of all fossils which chart the course of evolution throughout time. It is one of the most significant pieces of supporting evidence to the idea that evolution is a rational and real explanation for the origin of the diversity of species. In general, it refers to all fossils that have been excavated, examined and characterized so far by hundreds if not thousands of hardworking scientists and palaeontologists over the last few hundred years. The fossil record is used as evidence for many of things, including common descent, mass extinctions, the Cambrian explosion and the fact that the Earth is considerably older than 6000 years.
At the same time, the fossil record is yet another thing which shows that creationist ideas are untenable. Creationists like to claim that there are no transitional fossils (fossils that fill the "gaps" between one species and another) in the fossil record. This is probably because if one was presented to them, they would point out that there are two new gaps either side of it for which no transitional forms exist. Either that, or they are convinced that the fossil record is a hoax. But to scientists working on real research, the fossil record is a vital piece of evidence to study.
Due to the relative rarity of fossil formation (with factors such as time, location and pure luck governing what we can see), and the natural destruction of rock that holds fossils, it is unlikely that the fossil record will ever be 100% complete and comprehensive. Creationists don't understand this, and think that the absence of a single fossil disproves evolution outright. Scientists, and palaeontologists do understand how fossilisation works; noting that we can derive a lot of information from what we do have rather than dwelling on the information that we don't have/haven't found yet.