"Race" has no basis in biology, but "race" has very real social implications. In other words, the concept of race in understanding human biological variation has been basically abandoned by biological anthropologists (because it's scientifically worthless), but the concept of race is still used to categorize people (i.e. profiling, stereotyping, etc.).
This does not, however, imply there are no real biological differences between people - of course not - unless you're talking about twins, who are basically clones with no or very, very few genetic differences. In fact, differences/variations between people are very interesting as they give clues to our evolutionary origins, and the history of groups of modern people (i.e. were the first Americans from Asia? Where'd the Hawaiians come from?). They also make us realize how closely we're all related - the differences aren't very much even if they make skin color & other external appearances seem pronounced.
Basically, race implies that there are sharp distinctions or gradations between groups of people when it comes to their appearances/characters. This is simply not the case. Human variation is always
clinal in nature, with various characters present where they have been selected for, or simply as a result of natural population history (in the case of non-functional or marginally functional characters).
For example, sickle cell anemia is thought of as a 'black' disease in the US, but the gene responsible for sickle cell anemia is also found throughout the Mediterranean, in people who are not at all 'black' - it's just that in America the majority of folks with sickle cell disease are black. As another example, a single nucleotide substitution is responsible for the two major classes of ear wax type in people (haha, yes, this has been researched) - 'black' & 'white' people tend to have one type (moist - it fluoresces under uv light!), while 'yellow' & 'red' people tend to have another (dryer, flakier). I suppose if our overriding concern was with ear wax rather than skin color, Europeans & Africans would've been lumped together. The groups you make really just depend on the traits you're looking at.
Gould's "Mismeasure of Man" is an excellent and very accessible book on the subject. If you're really curious, use Google Scholar to look up articles on the subject by C. Loring Brace, who along with Ashley Montagu, really championed a non-racist approach to studying human biology. He recently wrote a fine book on the subject,
Race is a Four-Letter Word, which I highly recommend. Stephen Gould's
Mismeasure of Man is more accessible, and probably easier to find. It's also a very thorough refutation of the notion of race as biologically valid - it also destroys the validity of IQ tests (reification much?) & refuted the arguments made in "The Bell Curve" before that book was even written. Jared Diamond's
Guns, Germs, & Steel also makes it impossible to believe that the achievements of European & Asian societies have anything to do with inherent biological superiority.
For excellent overviews of the genetics of human diversity, check the homepage of the International HapMap Project (
www.hapmap.org).
Particularly relevant HapMap link. If you're really interested in this subject, feel free to PM me.