Evidence that apes and humans share a common ancestor comes from many fields. Below will be only a partial list, to give you an idea of just how overwhelming it is.
(1) Fossil intermediates morphologically intermediate between modern apes and modern humans.
Exhibit A, below, taken from
the TalkOrigins 29 'Evidences' for Evolution FAQ:
Fossil hominid skulls. The first, A, is a modern chimpanzee. The last, N, is a modern human. Those in-between are some of the more important fossils we've found, arranged
in linear order. Notice how the earliest fossils are more and more ape-like, whereas the later ones are more and more human-like. The ones in-between more or less completely blur the line between human and ape, possessing excellent combinations of primitive and derived traits. Especially note the less elongated jawline and increasing brain size from earliest to latest skulls.
The theory of common ancestry imposes
very stringent restrictions on what we should find and (and not find!) in the fossil record. Since we see a linear change from less human to more human, the theory is spectacularly corroborated. Since we
don't ever see any fossils that are totally prohibited by evolution, such as apes with eagle eyes or other mix-and-match uses of the best designs available, evolution is doubly corroborated.
Both tests (as well as others) are an empirical experiment that tests evolution's validity. It passes with flying colors.
(2) Identical pseudogenes in apes and humans that corroborate the powerful fossil evidence.
Taken from the
TalkOrigins Plagiarized Errors and Molecular Genetics FAQ:
Pseudogenes are formerly active genes that have been been the victims of crippling mutations and no longer express protein. Our genomes are littered with them, from the GLO pseudogene that produces vitamin C to dozens that control the olfactory systems in other mammals.
Normally, genes are preserved quite well by natural selection, and individuals with crippled genes are filtered out of the population. Occasionally, however, a gene with no mission-critical function will get deactivated and the organism can get along equally well without it. For example, a gene coding for the production of a protein that's abundant in the animal's diet anyway is irrelevant to its reproductive fitness.
Crippling mutations come in literally dozens of varieties: parts of the gene may become overwritten with another; parts of the gene may be cut out; random detritus can be spliced in, breaking its function; two genes can be merged together; parts of the gene can be shuffled with other parts; etc. We know they occur because we observe them in the lab and can even artificially induce them.
The causes of mutations range from errant cosmic rays to toxins in the environment. The common adage of "no two snowflakes are alike" applies even more here, as the odds of getting the same crippling mutation twice are somewhere in the astronomical range.
This leads us back to the newly formed pseudogene. As long as it bears no selection disadvantage, random genetic drift can establish the damaged gene in a small population. Should that population be successful and displace the others, the gene will be established in the entire species as a molecular vestige.
Any two offshoot species will hence be identifiable as related if they share the same crippled mutations in damaged genes. Does this apply to humans and apes?
The answer is a resounding "Yes!". Examples are listed below.
Primates, unlike all other mammals (with the exception of guinea pigs), cannot synthesize Vitamin C. In days long past, this led to tragic outbreaks of scurvy on seafaring voyages. Using the predictions of evolution, scientists hypothesized that the gene for vitamin C production would be found in humans as well, despite our not being able to produce it.
Lo and behold, a GLO (ascorbic acid pseudogene) was identified in humans at
exactly the same spots other mammals have functional vitamin C genes. What's more, the other great apes (chimps, gorillas and orangutans)
had an identical broken pseudogene!
The common ancestor of apes and humans lived in a fruit-rich environment and had no need to synthesize their own vitamin C, making the loss of that gene entirely neutral. Guinea pigs also have a damaged GLO pseudogene, but the mutation that crippled it is
different, as expected if it was an independent occurence.
Other occurences of shared pseudogenes include the one coding for Urate Oxydase, which make our species vulnerable to gout, and dozens of them that code for powerful smell in other animals but are crippled in humans, but one should suffice for now.
Observations in the area of pseudogenes that would falsify evolution include finding the same pseudogene in humans and dogs but
not apes; since apes and humans share a closer ancestry than dogs and humans, any pseudogenes found in dogs and humans MUST be found in humans and apes because they belonged to the common ancestor of the latter. Hence, the theory of evolution passes this series of empirical experiments as well.
(3) Atavisms. Lost traits indicative of our species' history that occasionally resurface in modern individuals and again corroborate hypothesized evolutionary relationships.
Taken from
here:
Exhibit B, X-ray image of an atavistic
human tail. The subject is a 6-year old girl.
Note: pseudotails are an entirely different phenomenon and unrelated to real human tails. Pseudotails are merely abnormal protrusions of skin and muscle in the sacral region, whereas some
real human tails have nerve ganglia, cartilage, additional vertebrae, and can be consciously moved and contracted. In addition, cases have been reported where they've been inherited in families.
Human fetuses are frequently covered in a minute fur (called lanugo) that becomes reabsorbed shortly before birth. Rarely, this feature is retained, resulting in what is known as Werewolf Syndrome by the press - a human covered in thick, animal-like fur.
This evidence more or less speaks for itself, but I'll offer a few additional comments.
Atavisms are another empirical test for evolution's validity. The theory predicts only those atavisms that reflect traits in ancestral species will be found. This, for example, means that finding wing atavisms on a human, gills on a whale, fur on a fish, or some such, immediately and uncontrovertably falsifies evolution.
On the other hand, finding excessive fur on a human, developed hind legs in whales or snakes, or other atavisms
totally consistent with their hypothesized evolutionary history is a strong confirmation.
Some of the evidence for chimp-human common ancestry. 
Linear Mode
