There is a common misunderstanding about evolution which must be addressed first:
Evolution is a fact, because we can see it happening. For example, every time a new strain of the influenza virus appears, it is genetically different from previous strains. This genetic change *is* evolution, by definition, simply because that's what we call it. That's why evolution is a fact. Even creationists accept this. When people say that evolution is 'only a theory', what they're really thinking about is common descent.
'Common descent' is the theory that all living and extinct organisms on Earth are related by reproduction, in the same way that members of a human family are related to each other by reproduction. Although we know that evolution is a fact, because we observe it happening every day, this in itself does not prove that *common descent* is true. For that, we need other evidence.
People sometimes question the validity of evidence for evolution, saying "How can you know this is true, if you weren't there to see it?" It is certainly true that humans were not around for most of the last 4 billion years or so, to witness first hand the origin of living organisms. However, many crimes also go unwitnessed, and we still feel confident enough about deducing what happened from the evidence available later, that a person's life or freedom can be spared or forfeited in a court of law on the strength of it. So, the lack of eyewitness testimony is not a valid objection to the argument for evolution.
So, here is some of the evidence for evolution and common descent:
o Evolution is observed to occur today - e.g. new strains of viruses. Even creationists don't deny this.
o There no reason to doubt that evolution has always occurred.
o There is no evidence for any mechanism limiting genetic change.
o The fossil record show constant change throughout the history of life, as would necessarily be the case if evolution was always occurring.
o The fossil record shows species arising, diversifying and then going extinct, as would be expected if evolution was occurring.
o The fossil record shows more differences from today's flora and fauna the further back we look in time, as would be expected if evolution always occurred.
o Geographically isolated places (e.g. islands) have many unique species, as would be expected if ancestral species arrived and evolved there.
o Isolated islands almost always have plants and birds but no indigenous amphibians or large mammals, even though the habitat would support them. This is exactly what you would expect if species could only arrive on islands by air or sea, rather than being put in place by a creator, who could have introduced any species anywhere he liked.
o The fossil record shows that new species appear geographically close to similar species, rather than in arbitrary locations, which is what you would expect if one evolved from the other.
o The fossil record shows that similar species inhabiting different environments tend to be close geographically, which is what you would expect if one ancestral species evolved and diversified into different habitats in one area - the pineapple family of species are a good example, being found in different habitats in the American tropics but not in other tropical parts of the world.
o When the fossil record is compared with other, independent scientific theories, e.g. plate tectonics, the results are consistent with evolution having taken place - e.g. species in South America and Africa are more similar the further back we look in time, which is what would be expected after the two continents were separated by plate tectonics and species on each side evolved independently since that time.
o Evolution predicts that biogeography must be consistent with a common ancestor, and this is what we find - e.g. marsupials are found in Australia and South America, so the earliest marsupial fossils must be found in rock strata formed before these landmasses separated, and this is indeed what we find. Moreover, evolutionary theory predicted marsupial fossils in Antarctica for the same reason, even though no marsupials live in Antarctica today, and that is exactly what was found.
o Many 'transitional fossils' have been found, illustrating the common origins of different groups of organisms.
o We never find fossils in rocks which are older than their presumed ancestors - the classic example is that we never find rabbits in the Pre-Cambrian (because they could not exist before the earliest mammals, which were the ancestors of rabbits).
o Organisms never breed offspring of a different group, e.g. a dog born from a cat. This is to be expected if evolution is true, since it would be inconceivable, according to the theory of evolution, for many hundreds of thousands of genetic changes to happen all at once and produce a viable organism - especially not matching one that already exists.
o Different continents often have different species in the same kind of habitat - e.g. the ostrich, emu and rhea, which are all large flightless birds living on grassy plains but on different continents. This is what would be expected if they each evolved independently to suit that particular habitat.
o Evolution predicts that genes and their products diverge as species evolve, accumulating greater differences between more widely separated species, and this is exactly what we see.
o Living and extinct species fit a statistically valid phylogenetic tree - Like a family tree for people, but immensely larger. This would be necessarily true if they are all related by common descent, but not expected if they are independently designed and created.
o The chirality of DNA, RNA and proteins is the same in all living organisms. This is to be expected if they all share a common ancestor, but not necessarily otherwise.
o All living organisms use DNA and RNA, never a different genetic material, which would necessarily be true if they all evolved from a common ancestor which itself used DNA and RNA.
o All living organisms use only 4 nucleosides out of hundreds of possible molecules, which would be expected if they all shared a common ancestor.
o All living organisms use the same 22 amino acids out of 390 possible choices, as would be expected if they shared a common ancestor.
o The genetic code is universal, with only minor differences, and those differences only between major groups e.g. plants and vertebrates. This is to be expected if they all evolved from a common ancestor, since any fundamental change in the code of an existing species would be extremely unlikely to produce viable offspring. However, there would be no particular reason to expect this if all species were created independently.
o All known species share the same energy storage molecule, ATP, as would be expected if they share a common ancestor.
o Vestigial features are common in living organisms, e.g. wings in flightless species, eyes in species which live in permanent darkness, a pelvis in pythons, vestigial legs beneath the skin in lizards, sexual organs in organisms which reproduce asexually and the coccyx in humans. This would be expected if species are the product of unthinking, undirected evolution, but not if they are the product of an intelligent designer.
o Living organisms display numerous atavisms, e.g. children born with tails, which is what we would expect if species still carry the genes which coded for the fully formed and functional organ in an ancestral species from which they evolved.
o We see fundamentally very different species of organisms with similar features in the same environment, e.g. sharks and whales both occupying the same environmental niche. This is to be expected if they evolved through widely separated evolutionary pathways to suit the same habitat.
o Observation of the embryological development of animals leads to testable predictions of their evolutionary origins.
o Parahomology is the similarity in structure of living organisms despite differences in function. Evolution accounts for parahomology - e.g. the ear bones of mammals can be seen to have evolved from the jaw bones of reptiles.
o Living organisms have many suboptimal functions and structures, e.g. the blind spot in the mammalian eye and the use of the same tube for both ingestion and respiration (the throat). This is to be expected if organisms evolved with no direction or foresight, but not if they were intelligently designed.
o Genetic sub-optimality is explained by evolution, e.g. why one single-celled organism has 45 times as much genetic information as almost identical species, and 3 times more DNA than humans. There would be no reason for a designer to make it this way.
o Molecular sequences of ubiquitous genes (genes found in virtually all living organisms) have high functional redundancy. This means that where a gene performs the same function in all known species, it always has the same or closely similar sequence in every species, despite the fact that a very large number of other possible sequences would be functionally equivalent. There is no particular reason for this unless all species are related by common descent.
o Evolution predicts that evolutionary change in the fossil record should be broadly consistent with the rate of mutations observed in species today, and this is what we find. For example, the evolutionary divergence of chimpanzees and humans from a common ancestor of around 6 million years ago gives an estimate of 2 x 10^-8 base substitutions per site per year in those organisms. Observed rates are between 1 and 5 x 10^-8 per year, a very good match with the prediction of evolutionary theory.
o Observed rates of mutation easily account for the observed differences between species as diverse as mice, chimpanzees and humans in the time frames indicated by the fossil record - In other words, consistent with evolution.
Every single one of these different, independent and mutually supporting forms of evidence for evolution is supported by dozens, hundreds or thousands of individual observations. They all make sense according to evolutionary theory. None of them make any particular sense if you suppose that living organisms are *not* related by common descent. Moreover, unlike religious beliefs such as 'intelligent design' they are all falsifiable, and are therefore valid forms of scientific evidence.
Finally, there is the philosophical argument: It is utterly inconceivable that the complexity and organisation inherent in living organisms could just exist from nothing, and this is the fatal flaw of any concept of a designer: If you argue that life requires a designer, then the designer would have to already exist, with the complexity and intelligence necessary to design living organisms... but then you have contradicted your argument by asserting that complexity does not in fact need a designer – You’re saying it *can* just exist without a designer. Any argument which posits an inevident designer only raises a bigger question than it answers, and ends up with the logical fallacies of either infinite regression or disproving its own premise.
The only alternative, then, is that complexity and organisation arise from simplicity and chaos by the operation of unthinking, undirected natural processes. In the case of living organisms, this means evolution.
ADDENDUM: 'Frankly', I wrote it myself.