The fossil record has an abundance of now-extinct life forms exhibiting characteristics intermediate between taxonomic groups that are now separate, or combining features of two groups in which the features are now consistently separate. We call them transitional forms, because they look as if they were a transitional stage in the evolution of one group of organisms into another group.
At this point of the discussion, it does not matter whether the transition really occurred. The immediate issue is whether there is evidence for a transition -- facts that would be explained by a transition if the transition had occurred. We have fossils that look transitional. Their existence is a fact, and therefore they count as evidence. If common descent should be disproved by other evidence, then we will have to conclude that the transitional appearance is illusory. Pending such disproof, though, we justifiably say that since they look transitional, we may infer at least the possibility that a transition did occur.
For some examples, in recent years numerous fossils have been found with both avian and reptilian characteristics. Farther back chronologically in the fossil record is a proliferation of fossils with a combination of reptilian and mammalian characteristics. Fossils of whale-like creatures with legs have been found, and others have been found that look like sea cows with legs. Still older fossils look like creatures intermediate between fish and amphibians and between amphibians and reptiles.
The chronology of these fossils is as significant as their existence. The more distantly related the modern groups are, the older the apparently ancestral fossils are. Fossils from roughly 40 million years ago show a creature called cynidictus that looked somewhat bear-like and somewhat wolf-like but not quite like either a wolf or a bear. Much later, in the neighborhood of 5 million years ago, there was an animal called Protursus simpsoni, which looked more like a bear than a dog but not quite like modern bears. Meanwhile there had appeared Cynodesmus, which was not exactly a wolf or a fox but looked more like either of them than like a bear.
This happens consistently as more fossils get discovered. Fossils that look intermediate between carnivores and artiodactyls are older than fossils that are clearly either carnivore or artiodactyl.
Of course not all descendant groups replace their predecessor groups. We must not forget the Chihuahuas and the wolves.
Thus, fossils that look intermediate between mammals and reptiles are older than fossils that are unambiguously mammalian, although some reptile fossils are even older still. Fossils that look intermediate between reptiles and amphibians are older than unambiguously reptilian fossils -- and also older than reptile-mammal intermediates -- but there are amphibian fossils that are even older. And, fossils that look intermediate between fish and amphibians are older than amphibian-reptile intermediates but not as old as some fish fossils.
Let's look at it another way, considering only vertebrate fossils.
At the beginning of the vertebrate fossil record, we find only creatures that are certainly fish. There are no amphibians, reptiles, mammals, or anything like a combination. The oldest vertebrate fossils are pure fish, nothing else.
A little later, we find, in addition to fish, creatures with a mix of fish and amphibian characteristics. We do not, however, find any reptiles or mammals, and we do not find any creatures that are purely amphibian.
A little later yet, though, we do find creatures that certainly are amphibians -- but no reptiles or mammals show up yet. This is what the fossil record implies. The oldest fish fossils are older than fossils that look intermediate between fish and amphibians, and the oldest intermediate forms are older than the oldest fossils that are unambiguously amphibians.
And so it goes for the reptiles and mammals. The oldest unambiguously amphibian fossils are older than the oldest fossils that seem intermediate between amphibians and reptiles, and the oldest of those intermediates is older than the oldest unambiguously reptilian fossil. It is likewise for reptile-mammal intermediates and unambiguous mammals.
The fossil record, in summary, is consistent with a supposition that some fish became amphibians, some amphibians became reptiles, and some reptiles became mammals. There are intermediates for all of them, and the intermediates are in the right chronological places.
If this ordering had never been observed -- if all taxonomic groups had been found in all geological strata -- it is unlikely that the scientific community would ever have embraced any theory of common descent. Even now, the theory could be seriously undermined by the discovery of fossils out of sequence. The discovery of a human fossil, or a chimpanzee fossil, or a horse fossil, or a wolf fossil, alongside a dinosaur fossil in rocks known to be 100 million years old would be very problematic. The discovery of any vertebrate fossil in Precambrian rocks would be very hard to reconcile with the theory of common descent.
For anyone wanting detailed information, Kathleen Hunt's Transitional Vertebrate Fossils FAQ is a must-read.
Next: The facts: Vestigial structures