Five Proofs of Evolution
1. The universal genetic code. All cells on Earth, from our white blood cells, to simple bacteria, to cells in the leaves of trees, are capable of reading any piece of DNA from any life form on Earth. This is very strong evidence for a common ancestor from which all life descended.
2. The fossil record. The fossil record shows that the simplest fossils will be found in the oldest rocks, and it can also show a smooth and gradual transition from one form of life to another.
Please watch this video for an excellent demonstration of fossils transitioning from simple life to complex vertebrates.
3. Genetic commonalities. Human beings have approximately 96% of genes in common with chimpanzees, about 90% of genes in common with cats (source), 80% with cows (source), 75% with mice (source), and so on. This does not prove that we evolved from chimpanzees or cats, though, only that we shared a common ancestor in the past. And the amount of difference between our genomes corresponds to how long ago our genetic lines diverged.
4. Common traits in embryos. Humans, dogs, snakes, fish, monkeys, eels (and many more life forms) are all considered "chordates" because we belong to the phylum Chordata. One of the features of this phylum is that, as embryos, all these life forms have gill slits, tails, and specific anatomical structures involving the spine. For humans (and other non-fish) the gill slits reform into the bones of the ear and jaw at a later stage in development. But, initially, all chordate embryos strongly resemble each other.
In fact, pig embryos are often dissected in biology classes because of how similar they look to human embryos. These common characteristics could only be possible if all members of the phylum Chordata descended from a common ancestor.
5. Bacterial resistance to antibiotics. Bacteria colonies can only build up a resistance to antibiotics through evolution. It is important to note that in every colony of bacteria, there are a tiny few individuals which are naturally resistant to certain antibiotics. This is because of the random nature of mutations.
When an antibiotic is applied, the initial innoculation will kill most bacteria, leaving behind only those few cells which happen to have the mutations necessary to resist the antibiotics. In subsequent generations, the resistant bacteria reproduce, forming a new colony where every member is resistant to the antibiotic. This is natural selection in action. The antibiotic is "selecting" for organisms which are resistant, and killing any that are not.