The five evidences that support the theory of evolution are: Fossils, Anatomy, Embryology, Molecular Biology (or Biochemistry), and Biogeography.
Fossils show the physical change in a species over time, suggesting evidence for evolution. For an example, we can trace the origins of dolphins and whales all the way to a four-legged, dog-like hoofed mammal from about 50 mya. These mammals were fish eaters and so decided to spend a lot of time in the water. In fact, they spent so much time in the water, that in a few million years, they evolved a longer body and webbed feet. Then another few million years passed and they swapped their webbed feet for flippers and swapped their tail of a fin. Eventually, we get a fish-like mammal that once walked on land.
For anatomy, or structures, there's three types of structure comparisons: Homologous structures, Vestigial structures, and Analogous structures (or Convergent Evolution). Homologous structures show similarities between the structures between species, but serve different purposes. For an example, all mammals share common features within their arms, such as having 5 fingers. However, they serve different purposes, from whale fins for swimming, to bat wings for flying, to human hands for grabbing and for using tools.
Vestigial structures were structures that once served a purpose and have been reduced in size/power to the point where it's useless. For an example, large theropods such as the T. Rex spent so much time evolving more jaw muscles and biting power, that they sacrificed their arm muscles for such power. This explains why their arms were so short.
Another example would be ratites, flightless birds such as ostriches, emus, and cassowaries (penguins and kakapos are from a different group). They spent so much time evolving powerful legs as well as evolving in size, that there was no need to fly when they could just run with their improved legs. And also the fact that they're too heavy to even fly anyways.
Analogous structures is the similar functions of structures between species, but have separately evolved these functions. For an example, the wings from insects, birds, bats, and pterosaurs are very similar in shape and can be used to fly, but they've evolved separately. For an example, birds use feathers to fly while bats and pterosaurs use/used (pterosaurs are extinct) membrane between their bones to fly. Although they do have strikingly similar evolutionary histories, with their ancestors starting off as tree dwellers, they've evolved their flight abilities separately during different earth periods.
Embryology is evidence for evolution by using embryos from different species to suggest a common ancestor. For an example, we can prove that we're more related to whales than to fish by using embryos. The fish embryo will evolve gills while the embryos of humans and whales will not, since we do not breath by extracting oxygen from water.
Molecular biology is a more recent way of providing evidence for evolution. By taking the DNA from cells from different species, we can get an estimate of how related one species is to another. The more similar amino acid sequences are between species, the more recent their common ancestor is. For an example, by comparing amino acid sequences between humans and chimpanzees, we get an estimated amount that we once shared a common ancestor seven million years ago. Without molecular biology, we would still be digging for fossils to find that number.
Biogeography is the geographic distribution of species on earth that can be explained by evolution. For an example, we can tell that Australia separated first from Pangea rather than Asia or North America since Australia doesn't have bears while the Asia and North America do, with China having pandas and North America having brown and black bears. And also with the fact that Australia is the only continent with monotremes (egg-laying mammals like the platypus).
Similar to biogeography, there's also direct observation, which is evolution that has happened in our own eyes. An example would be the underground London mosquito. Mosquitoes usually target birds like pigeons. But after an underground tunnel was built in London, some mosquitoes managed to get inside and instead of birds, they targeted, mice, rats, and of course, humans. After only a few hundred years, a mosquito from the underground tried to mate with another mosquito from land and they could not, simply because they were different species. This shows that their new choice of feeding on mammalian blood has actually altered their DNA and created a new species, during the existence of humanity!