This will simply be a brief look at modern Christian explanations for the fossil record. If any of these fit your view, then feel free to expand on what I write and explain why you believe it. If I have missed any, then please present the views I have missed.
1. Fossils can only tell us that something died. This sort of statement is often used in order not to address the fossil record or to dismiss any succession perceived in it.
2. Fossils are a test of faith. A view which allows for dismissal of the evidence.
3. Fossils were planted by Satan. This is similar to 2.
4. Fossils are fake. A rare view which invokes global conspiracy as the explanation.
None of the above views are serious attempts to engage with the evidence and so will not be addressed in any more detail.
Young Earth Creationist views:
The following four views are not necessarily mutually exclusive and all invoke a global flood.
1. Hydrological sorting. This is the view that the fossil record can be explained by organisms being sorted by density and other factors rather than evolution.http://creationwiki.org/Hydrological_sorting
2. Differential escape. This explanation is used to account for the increase in complexity seen in the fossil record. It posits that smarter, faster animals were able to escape the flood for longer, while all the slower, dumber animals died first, giving the illusion of evolving complexity. It also tries to explain the abundance of marine fossils.
3. Habitat sorting/ecological zonation. In this explanation, differing environments are used to explain fossils found in different strata and locations. The reason we do not find trilobites and man together is because trilobites are marine and humans are terrestrial, for example.
4. The European flood model. This is more complex than the other YEC models, as it does not propose the flood as an explanation for the whole fossil record. Instead, the pre-flood/post-flood boundary is in Carboniferous strata. The fossil record shows a succession of pre-flood organisms which died (all were marine because the land was destroyed), through organisms which proliferated during the flood, followed by the progressive filling of the Earth after the flood receded. A proponent of this view is Paul Garner.
Favourite evidences used by the YEC groups include polystrate fossils and fossil lagerstatte (areas of exceptional preservation). Proponents include Duane Gish, Henry Morris, Kent Hovind and the palaeontologist Kurt Wise.
I have included this one for a laugh. I once met a guy on a different discussion board who is trying to raise funds to start a museum which will explain the fossil record his way:
The greater ancestor explanation: The fossil record shows devolution, where ancestors were bigger and better in the past and have been progressively getting worse. Favoured evidence includes large mammal fossils from throughout the Cenozoic and large insect fossils from the Carboniferous.
Old Earth Creationist views:
1. Gap theory: This explains the fossil record by considering it as almost a separate creation, before God wiped the slate clean and started afresh with man and other animals. The fossil record is therefore the remnants of a failed creation.
2. Concordism and day-age: This view accepts the age of the Earth and instead suggests that there were many stages of creation and that Genesis matches what the fossils show. No large scale evolution was involved, just God stepping in to make created kinds.
Proponents of these views try to take the fossil record as a whole, though tend not to engage with it much. A prominent proponent is Hugh Ross.
1. The patterns in the fossil record are real, but not explained by evolution. This is simply a rejection of evolution as an explanation for the whole fossil record. ID theorists differ in the degree to which God has stepped in during geological history.
Evidence given often includes the Cambrian explosion as a rapid source of new "information", and also examples such as whale evolution are used to claim that evolution would be too slow. Proponents include Michael Behe and Stephen Meyer.
Theistic evolution views:
1. Evolution as cosmic warfare/modified gap theory: The fossil record is explained by cosmic warfare between Satan and good, corrupting the creation and causing mass mortality. It accepts a literal Genesis with a gap in Genesis 1.http://www.gregboyd.org/blog/evolution-as-cosmic-warfare/
2. Teleological evolution: Evolution is the explanation for the fossil record, but it also shows that there is an inevitable progression in complexity, which points to God. Proponents of this view, such as palaeontologist Simon Conway Morris, use palaeontological evidence of convergence in evolution as support, combined with constraints in nature. http://www.mapoflife.org/
3. Generic theistic evolution: Simply accepts evolution as the explanation for the fossil record, based on the evidence for an increase in the spread of complexity and faunal succession, along with transitional forms and other lines of fossil data.
Proponents of theistic evolution include Kenneth Miller, John Polkinghorne, and palaeontologists Bob Bakker and Mary Schweitzer.