Lately the Russian Federation is showing increasing involvement in the Syrian civil war.
What could Russia want to accomplish by supporting Assad?
The alliance between Syria and Russia dated back to the Cold War when they both became enemies of the United States. After the Iranian revolution and the peace between Egypt and Israel, Iran became a natural ally of Syria and the center of a Shiite block against USA, Israel and Iraq.
Given this durable alliance Russia has developed many interests in Syria. One of them is navy base in Tartus, which is quite modest but is the only Russian base in the Mediterranean Sea. A more practical reason is money: Russian banks have supported the financial system of Syria since the start of the civil war and its companies have invested 20 billion dollars in the extractive industry1.
Russia has another specific reason to support Assad in the current civil war: islamic terrorism, because it is also an internal Russian problem. They believe that the Assad government and the Kurds are the only reliable forces that are fighting against ISIS2.
[1] Siria: Stivali russi per Assad (it)
[2] Putin’s U.N. General Assembly speech
Qatar strives for constructing a pipe via this region to Turkey to supply EU with its own oil and gas from the giant North Gas Field. Syria blocks the way for two reasons:
1) Qatar's pipe would severely drop the EU dependency from Russian energy sources and the EU will be more independent from Moscow. 2) Qatar's pipe would harm Iranian interest in supplying EU with its own oil and gas.
That is why Russia supported sanctions-free Iran and strongly opposes the idea to overthrow Assad in Syria, the end of his personal reign or a sort of division of the solid state into several separate ones based on religious/ethnic majority/minority would open the way to Qatar's pipe and shrink Gazprom European influence. Saudi victory over Iran as a byproduct, by the way.
However, freezing the conflict for several years will block Qatar's project and then Iran's energy sources will flow freely to EU via Turkey ending Qatar's efforts finally forever. As Iran is a puppet of Russia, the EU dependency will only strengthen and the US influence in Europe will wither. A catastrophe.
In addition to the points mentioned by Gabriele, one also has to consider the longstanding Russian and Chinese doctrine of respecting the sovereignty of nations. The only exceptions allowed is when Article 51 of the UN charter can be invoked for self-defense or if the UNSC has passed a resolution under Chapter 7 authorizing an intervention. So, Russia does not accept the Western logic that under certain other circumstances one can also intervene, declare that a president of a nation should no longer be recognized as such, etc..
It is then that Russia does not take the actions that the West has taken w.r.t. Syria (e.g. oil embargo, weapons embargo etc.), that makes it look like Russia has actively supported the Assad regime all along and is now doubling down on that. But while Russia has now started to engage a bit more with Syria to protect its interests in Latakia, it's this false perspective of the Russian engagement with Syria that makes it look like a lot more than there is to it.
If Russia were to send a few divisions of its army to Syria then that would be a different matter. But a few dozen aircraft there that have not launched any attacks, that's not a big deal. They may never see any action, Putin could just ask Obama to tell the rebels in Idlib to stay out of Latakia or else they'll be attacked there.
One can compare this to the longstanding US policy of "supporting Israel". Here one also has to make the remark that Israel being a sovereign nation can have military ties with any other nation of its choosing. When the US sells arms to Israel, that in itself does not per se constitute "support for Israel" in all of the conflict Israel is engaged in (the US is critical of Israel on some issues). But because many do take that perspective, that causes any other more active support for Israel in a conflict to be construed as much more than it actually is.
So, we shouldn't be surprised if Russia ends up supporting a political transition leading to Assad's departure as part of a negotiated peace plan with all the parties involved, despite the current military moves in the region. The bottom line is that what Russia is doing isn't really "support for Assad", it's much more continuing the status quo w.r.t. to Syrian and Russian relations, and these are motivated by the points mentioned in Gabriele's answer. Russia won't change this just because the West decided that time is up for Assad.
Let me clarify the issue about the "doctrine of respecting the sovereignty of nations" which has been raised in the comment here. The whole point of this doctrine is to prevent chaos. While the Russians have not always strictly obeyed this principle themselves, the general idea is that a bad leadership is better than undermining it and getting into a chaotic situation. The Russian will point to e.g. Libya and Iraq to point out that getting rid of bad dictators isn't always the best way forward.
So, you can't argue that because Russia is supporting Assad, that Russia must have some huge stakes in Assad remaining in power. This reasoning would only be valid if you don't support the Russian doctrine about intervention. In that case, the threshold needed to support the overthrow of a regime is much lower. And if that support would then not be forthcoming, then one has to ask what are the enormous stakes that country has to prevent them from giving their support here.
