NB: I'm posting about Civ 4. I gather this thread has been merged from threads in other forums, and presumably this explains why people keep going on about things which aren't in Civ 4, like having Cleopatra as a leader.
Regarding race, it depends on exactly what you're looking for. As mentioned above, the UI isn't very multicultural - the specialist icons are all white men. With the addition of the Beyond The Sword expansion pack, the military units of every civ are supposed to look correct for that particular culture (and for what it's worth, I personally hate this feature and will mod it out of my copy when I get the time - it makes it very difficult to recognise the different types of units). There are leaders from virtually all parts of the world, and naturally they are supposed to be representative of their respective cultures/ethnicities (but read the archives of this forum and you'll find lots of pointless debates about the Viking leader having a horned helmet). You may notice some other UI race issues - I'm not particularly attuned to noticing these things.
In terms of actual game mechanics, race mostly isn't handled. Insofar as it does get handled, it's mostly in the form of "culture". Every city produces culture belonging to its controlling civilisation. The actual amount of culture produced will vary, depending on how you play. Culture spills out onto surrounding tiles, and so different cultures can overlap. The civilisation with the highest level of cultural influence over a tile gets to control that tile, so borders can fluctuate. In an actual city, the ethnic/cultural makeup of the population is determined by the levels of different civs' cultures influencing the tile on which the city is located. For example, if London is near to Tenochtitlan and Medina, it might have 85% English culture, 10% Aztec culture and 5% Arabian culture. For the most part, this has no effect on gameplay. However, if a city has less than 50% of its culture from its controlling civ, there can be unhappiness penalties, and the city will potentially defect to the strongest influencing culture if the situation isn't remedied. Clashing cultures is a zero sum game. Cities never gain from having influence of multiple cultures; the only potential effects are happiness penalties and control flips. The religion system is a lot more sophisticated than the culture system, in this respect.
That's about it for game mechanics. As mentioned in earlier posts, many civics sound socially significant, but they don't interact with the issue of clashing cultures at all. There isn't a single civic which directly affects the (rare) happiness penalties from clashing cultures, nor the probability of control flips. The closest thing to a civic with interesting cross-cultural effects is Theocracy, which prevents rival religions from spreading into your empire. There are no civics which do this for culture itself. Basically, race just isn't really dealt with as an issue in the game.
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