Anatomical homologies are morphological or physiological similarities between different species of plants or animals. Comparative anatomy, which is the study of anatomical homologies, is the source of most traditional evidence for evolution and common descent. Anatomical homologies continue to provide many examples of deep relationships between species which are best or only explained through evolutionary theory when the similarities simply don't make sense from a functional perspective.
If species actually arose independently (naturally or through a divine act) each organism should have characteristics uniquely suited to its nature and environment. That is, an organism's anatomy would function in a manner most suited to its particular way of life. If species evolved, however, then their anatomy is limited by whatever their ancestors were able to provide. This means that they will lack some features which would be well-suited to how they live and they would have other features which aren't so helpful.
Perfect Creation vs. Imperfect Evolution
Although creationists like to talk about how life is "perfectly" designed, the fact is that we don't find this when we look around at the natural world. Instead, we find species of plants and animals which might do much better with anatomical features found in other species elsewhere and which are making do with anatomical features that appear to be related to other species, past or present. There are countless examples of these kinds of homologies.
One frequently cited example is the pentadactyl (five digit) limb of tetrapods (vertebrates with four limbs including amphibians, reptiles, birds and mammals). When you consider the vastly different functions of the various limbs of all of these creatures (grasping, walking, digging, flying, swimming, etc.) there is no functional reason for all of these limbs to have the same basic structure. Why do humans, cats, birds and whales all have the same basic five digit limb structure? (Note: adult birds actually have three digit limbs, but embryonically these digits develop from a five digit precursor.)
The only idea that makes sense is if all of these creatures developed from a common ancestor that happened to have five digit limbs. This idea is further supported if you examine the fossil evidence. Fossils from the Devonian time period, when tetrapods are thought to have developed, show examples of six, seven and eight digit limbs - so it is not as if there were some limitation to five digit limbs. Four limbed creatures with different numbers of digits on their limbs did exist. Again, the only explanation that makes any sense is that all of the tetrapods developed from a common ancestor that happened to have five digit limbs.
In many homologies, the similarity between species is not actively disadvantageous in any apparent way. It may not make sense from a functional perspective, but it doesn't appear to actually harm the organism. On the other hand, some homologies do indeed appear to be positively disadvantageous.
One example is a cranial nerve that goes from the brain to the larynx via a tube near the heart. In fish, this path is a direct route. What is interesting is that this nerve follows the same route in all species that have the homologous nerve. This means that in an animal like the giraffe, this nerve must make a ridiculous detour down the neck from the brain and then back up the neck to the larynx area.
So, the giraffe has to grow an extra 10-15 feet of nerve compared to a direct connection. This recurrent laryngeal nerve, as it is called, is clearly inefficient. It is easy to explain why the nerve takes this circuitous route if we accept that giraffes evolved from fish-like ancestors.
Another example would be the human knee. Rearward articulating knees are much better if a creature spends most of its time walking on the ground. Of course, forward articulating knees are great if you spend a lot of time climbing trees.
Rationalizing Imperfect Creations
Why giraffes and humans would have such poor configurations if they originated independently is something that remains for creationists to explain. The most common creationist rebuttal to homologies of any kind is frequently of the "God created all creatures according to some pattern which is why different species show similarities" variety.
Ignoring the point that we would have to consider God an extremely poor designer if this were the case, this explanation isn't really an explanation at all. If creationists are going to claim that some plan exists, it is up to them to explain the plan. To do otherwise is just an argument from ignorance and is equivalent to saying things are the way they are "just because."
Given the evidence, the evolutionary explanation makes more sense.