all members of Homo sapiens can mate with each other. every human in the world can have children with another human (of the opposite sex, of course

) we have almost exact DNA.
human DNA is VERY similar from place to place, because we evolved from a very small population chokepoint. about 70,000 years ago, it's estimated that less than 1,000 humans remained alive, perhaps as a result of climate and ecological changes which may have led to the last ice age (70,000 to 10,000 years ago). every human being alive today descends from those 1,000 or so people, so our DNA is very, very similar.
given our narrow DNA range, it's dangerous for us to mate with close relatives. most animals don't have this problem because their DNA range is much, much wider than that of humans. for example, generations and generations of cats mate with sisters, brothers, mothers, fathers, sons, and daughters, and no ill effects accrue.
at this point, humans are far enough distanced from chimps DNA wise that an offspring is not possible. however, there is evidence that early on, while the two species were closer together, there was interbreeding. this is not as weird as it sounds.
most closely-related species do in fact interbreed. for example, it's hard for dogs and wolves to mate, but they can do it and produce viable offspring. in fact, dogs and wolves are so closely related that dogs are probably more of a subspecies of wolf than their own species. this is not surprising, since human beings "genetically engineered" wolves by taming them and breeding them to produce smaller, less fierce wolves, which eventually became "dogs." but dogs are just wolves, and they are identical genetically (well, 99% identical). the gap between wolves and dogs is only 14,000 years (that is, dogs have been genetically different from wolves -- 1% -- for only 14,000 years. before that, they were wolves).
after a while, as they stick to themselves and less interbreeding ocurs (due to different habitats, habits, etc.), the DNA differences become distinct enough that hybrids will be born some or most of the time. for example, donkeys and horses are
closely related enough that they can mate and have offspring (mules) which can be hybrids but aren't necessarily. there are quite a few species that mate and make hybrids, indicating they are close in DNA terms, but not as close as dogs and wolves.
there are confirmed examples of polar bear/grizzly bear hybrids. this may be one of the ways in which new "species" split off (as the result of two existing species breeding).
at some point, the DNA is distinct enough that no insemination can occur (the sperm doesn't "recognize the egg, and vice versa). likewise, the appearance or behavior of the animals may change so much that they have no "desire" to mate with the other animal or mating is physically impossible.
great wikipedia articles on this:
http://en.wikipedia.org/wiki/Species
http://en.wikipedia.org/wiki/Hybrid
(please keep in mind that i'm just an amateur at this stuff. i read about it because it interests me, but i am by no means a trained scientist)