On the face of it, you wonder how anyone could vote against a resolution meant to try to prevent a dictator from murdering his own citizens. From a practical level, part of Russia's rationale for vetoing the UNSC resolution was simply driven by recognition of the deep, long-standing ties between their country and a loyal client state. It has been mentioned in media reports that Syria is a major buyer of Russian military exports; but Syria also hosts one of the few remaining foreign ports-of-call for the Russian Navy at the Mediterranean port of Tartus, without Syria, Russia would largely be shut out of the Middle East, a region in which the old Soviet Union enjoyed a fair level of influence. It's possible that any follow-on regime to Assad's might be willing to continue this historic relationship, but that is a risk that Russia does not want to take.
But the Russian/Chinese veto of the Syrian resolution was more than just a comment on UN policy towards Syria, it was also a symbolic line in the sand draw for the US-led “Western” community of nations that they were not going to be allowed to pick and choose which regimes stayed in power, at least as long as China and Russia had a say in the matter. Russia has been openly skeptical about last year's intervention in Libya, saying that the stated humanitarian mission was a cover story for the real goal of ousting a long-standing irritant to the West, Moammar Gadhafi. And when you look at the uneven way that the humanitarian military operation was conducted – with the US/NATO coalition overlooking rebel atrocities committed against pro-Gadhafi towns for example - there is something to this notion. Taking a look at the recent actions promoted by the United States, you can see a similar narrative shaping up against Iran (at least from the Russian/Chinese point-of-view), where the United States is pushing the global community to adopt a harsh sanctions regime targeting Iran's oil industry, meant to cripple the country economically by denying them revenue from their main export commodity.
That regime scheme is likely doomed to fail, in large part thanks to the Chinese – the largest buyer of Iranian oil exports – who are refusing to go along with the embargo. Part of the Chinese rationale, and also the reason cited by countries like India and Turkey, is that the Iranian sanctions lack the blessing of the United Nations. Saturday's vote makes it clear that such a blessing, either for more strict sanctions or ultimately military action against Iran, won't be coming thanks to the Russians and the Chinese. Both countries are concerned about American influence in their backyards – for Russia, the former Soviet Republics and Satellites in Eastern Europe and Central Asia; for the Chinese in the Pacific Rim and, again, Central Asia – changing the regime in Iran would be a real feather in the foreign policy cap of Pres. Barack Obama, a move he could parlay into gains in the Russian/Chinese spheres of influence. Russia and China therefore have a vested interest in making sure that such an event doesn't happen in Iran, Saturday's UN vote was just a small reminder of where things stand in this larger struggle.