1) The fossil record.
"The fossil record is incomplete! Where are the missing links?" ask creationists. Yes, the fossil record is "incomplete". The only way it could be "complete" would be if literally every single living thing had been fossilised after it died. That doesn't happen, because the process of fossilisation is incredibly unlikely, especially for land creatures.
But given how unlikely it is, the fossil record is amazingly good. Take any species of large vertebrate alive at the moment, and there's a good chance there will be fossils which could be its ancestor. In some cases there are lots.
For instance: Whales. We know whales are descended from land mammals. But for a long time, it wasn't clear which. Darwin thought their ancestors might be something like a bear; later evidence suggested it's probably a relative of cows and hippopotamuses. But there wasn't a fossil of a whale-ancestor on the brink of becoming aquatic.
And then, in 1994, a skeleton was found in Pakistan, of a 50-million-year-old animal which is now known as Ambulocetus natans. It's an ancestor of whales, but it has small hooves: It lived a largely but not exclusively aquatic life, like that of modern seals.
That's not the only whale ancestor. There's Pakicetus, a dog-sized predator which lived between 52 and 48 million years ago, and which appears to have been amphibious, perhaps spending a good portion of its time in the water, like a hippo, but still comfortable on land. Or, later, there's Aetiocetus. While Ambulocetus had nostrils on the tip of its snout, and modern whales have blowholes on the tops of their heads, Aetiocetus has nostrils half-way up its nose. It's a beautiful example of a fossil which shows how an earlier species evolved into a later one.
I've chosen whales: I could have chosen penguins, or turtles, or horses, or, of course, humans. Yes, a "missing link" has been found between humans and apes. In fact, several have. There is Sahelanthropus, an ape which lived around the time that humans and chimpanzees diverged. Then there are the Ardipithecus and Australopithecus ape-men. Then comes the arbitrary line where we start calling them humans: The genus Homo includes, among others, Homo habilis, Homo erectus, Homo ergaster, Homo heidelbergensis, Homo neandertalis and Homo sapiens, all of which have several fossil examples.
It's actually not very helpful to talk in terms of "transitional forms". All species are transitional. Humans will probably look very different, if we exist, in a million years' time, but we don't feel like a "transitional form" between Homo erectus and future humans. Instead it's worth talking about "transitional characteristics" between older species and more recent ones. Tiktaalik, which appears to be an early ancestor of amphibians, has lots of transitional characteristics between fish and amphibians: Its fins are limb-like and can support it out of water, and it has a lung-like organ. It lived about 375 million years ago, and we know about it from fossils found in Canada.
If anyone ever says to you that the fossil record is "incomplete", ask them how much more complete they would like it to be.
2) The spread of species.
People sometimes complain that evolution is "unfalsifiable". What they mean by that is that a scientific idea should make testable predictions, and that evolution, apparently, doesn't – so, if the theory of evolution is false, you can't prove it false.
That's nonsense. There are dozens – thousands – of testable predictions that the theory of evolution implies. Let's take a look at one subset: The geographical spread of species.
Marsupials are a group of mammals that give birth to their young at a much earlier stage than other mammals, and then carry them with them in a pouch on their bellies. The group includes kangaroos and wombats and opposums, among other creatures.
There's a confusing thing about them, though: They live on landmasses separated by thousands of miles of ocean. Most marsupials are found in Australia and New Guinea and other nearby islands. But 100 species or so are found in the Americas, mainly South America, with a few in Central and one in North America. They're not found in the Asian landmass which sort-of links the two, so they can't have walked, and they certainly can't have swum.
The theory of evolution predicts that there must have been some way that the ancestors of the Australasian and American marsupials made it into their respective continents, without having to swim across any oceans. In Darwin's time there was no available explanation.
But other lines of evidence put the common ancestor of modern marsupials at around 150 million years ago. And in recent decades, geologists have shown that back then South America and Australia were part of one huge supercontinent called Gondwana. Marsupials all lived in the same place, and the two groups were separated by the movement of tectonic plates.
The theory of evolution predicts that the geographical spread of species will be dictated by whether their ancestors could actually have made it there. Penguins could probably survive in the Arctic, but you don't find them there, because their ancestors lived south of the Equator. The common ancestor of Old World and New World monkeys lived before South America and Africa had split apart. To falsify the theory of evolution, you'd simply need to show that some species couldn't plausibly have made it from point A to point B. In the 156 years since Darwin published On the Origin of Species, that hasn't happened.
Evolution has to work on what already exists. If it's true, then we would expect to see that, for instance, body parts in one species can be mapped onto those of another, because they share an ancestor.
And that's exactly what we do find. Look at the human hand. It has five fingers, each with four bones, including the one in the body of the hand. And if you look at the forearms of all mammals, you'll see the same structure.
What's really surprising is that it's true even if the creature in question hasn't got "limbs" like ours. The flippers of whales and seals, and the wings of bats, have exactly the same pentadactyl (five-fingered) structure. And lizards and frogs have it too. It's because we all share an ancestor, a creature called Eusthenopteron, which lived about 385 million years ago.
It's not just limbs: You can see homologous structures in plants (the basic "leaf" structure has been co-opted for things as diverse as onion layers and Venus flytrap mouthparts). The ears of modern mammals include bones that are homologous to the bones of lizards' jaws. The mouth parts of insects are hugely diverse, depending on the lifestyle the insect has, but every one has the same basic structure. If evolution weren't true, there would be no reason to expect these signs of common descent.
The most striking evidence that all creatures share the same ancestor is this: They all share the same basic genetic code. The gene for an eye in a fruit fly will make an eye in a mouse. DNA is the language that all life talks in (unless you count viruses as alive, and even they use RNA, a simpler molecule, to hijack the DNA in other creatures' cells).
But the evidence from DNA is subtler than that. By comparing the genetic code of species, biologists have shown that more closely related creatures share more DNA. Humans share about 99% of their genetic material with chimpanzees, our closest relatives, but only about 96% with gorillas, our slightly more distant cousin. By comparison, we share about 35% of our genes with daffodils, our far, far more distant relatives.
As our understanding of genetics has improved, we've been able to use it to piece together great swaths of our evolutionary history. For example, the fact that modern humans interbred with Neanderthals was revealed by genetic analysis.
5) Convergent evolution.
The geographical spread of species is limited by their ancestry, as we've seen. But sometimes species separated by thousands of miles face similar challenges. A herbivore on the grass plains of North America would have the same sort of problems that a herbivore on the savannahs of sub-Saharan Africa would. If evolution were true, you would expect unrelated species to have evolved to come up with similar solutions.
And lo and behold, that's true. The American pronghorn looks and behaves much like an African antelope, but is not an antelope and is only very distantly related to them. Because it faced fast-moving predators on wide grass plains, it evolved long legs for sprinting and a nervous disposition, like its equivalents in Africa.
Aardvarks, anteaters, Australian echidnas, pangolins, and armadillos have all evolved to eat ants or termites, and have developed powerful arms for digging into the nest and long snouts with long sticky tongues to swipe their prey out of them. But these groups are incredibly distantly related; the last common ancestor of all five lived about 400 million years ago. For comparison, the most recent common ancestor of humans and pangolins lived less than 100 million years ago. The ant-eating specialities evolved independently, because they're a good way of solving a problem.
6) We've seen evolution happen, in real time.
Normally we think of evolution as something that happens over thousands or millions of years, and it often is. But there are plenty of examples of it happening in human timescales.
The most famous example is the peppered moth, which lives in forests in Britain and is camouflaged against tree bark. Up until the 19th century they were all white, but when the Industrial Revolution blackened the trees in British forests, the white colouring became much more visible. In 1811 a first dark specimen was recorded, a mutant. Against the dark trees they were much harder for predators to spot. By the end of the century it outnumbered the white ones. But as the heavily polluting industries in Britain fell away in the 20th century, and the forests became cleaner again, the white moth became more common.
Some creationists will tell you that the peppered moth is an example of "microevolution", and doesn't represent the "macroevolution" which would explain the creation of whole new species. In that case, point them to the apple maggot. Since the introduction of apples to North America, a whole new species of fly is steadily emerging. Before 1850, the ancestors of the apple maggot fed on hawthorn. Now, the subspecies of maggots which eat apples rarely eat hawthorn, and vice versa. The two are apparently in the early stages of speciation.
More importantly for humans, bacteria and viruses evolve much faster, because they go through so many generations so quickly. There are now dozens of kinds of microbe which are resistant to various drugs. Penicillin, the first breakthrough antibiotic, is largely useless these days, because so many bacteria are resistant to it. That is evolution in action: A bacterium which happened to have a mutation which protects it against an antibiotic will, in an environment where that antibiotic exists, have more offspring than its rivals.
7) Evolution is indeed a "theory". But "theory" doesn't mean "hunch". It is both a theory and a fact.
People who don't believe in evolution sometimes say it's "only a theory", because it's called the "theory of evolution". That's because, in everyday language, we use the word "theory" to mean something like "hypothesis" or "guess". But scientists use it to mean something much more specific. Here's how the American Association for the Advancement of Science puts it:
A scientific theory is a well-substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experiment. Such fact-supported theories are not "guesses" but reliable accounts of the real world. The theory of biological evolution is more than "just a theory." It is as factual an explanation of the universe as the atomic theory of matter or the germ theory of disease. Our understanding of gravity is still a work in progress. But the phenomenon of gravity, like evolution, is an accepted fact.