It's a bit of a long and complicated history, but I'll simplify it a bit. The two main factions were the Reds, who were Leftists, but mostly Socialists, and the Whites, who were more right wing and conservative, and comprised of everything from Monarchists, Military Dictators, and Republicans. Firstly, there were "Libertarian Socialists" if you count the opposition that was operating within the Bolshevik Party under people like Bukharin, that worked within the party but held a different and more "libertarian" (or as the Bolsheviks would call it, capitalist) view.

There were also the Left SRs who participated in the 1918 elections, and co-operated with the Bolsheviks until being expelled from the Congress of Soviets, and then leading an uprising against them in the territories ceded to Germany. As this revolt failed, a good number of them ended up going back the most prominent leftist party, the Bolsheviks.

Anarchists for sure did participate in the Russian Civil War, Nestor Makhno in Ukraine being the most prominent example, although the anarchists usually switched sides and co operated with both the reds and the whites. The green armies could possibly be counted too. They mostly were opposed to all sides and were organized peasant militias that basically sought to protect their communities from the war.

The White Army, as mentioned earlier, was essentially made up of the remnants of the former right wing politicians in Russia, and the people of the property and factory owning classes, like the nobility, bankers, and capitalists, that the Bolsheviks were working to overthrow and displace in the name of Socialist construction. While it was diverse in outlook, as mentioned earlier with being made up of monarchists, military dictators, and republicans, the most common trend was that of Nationalism, and general opposed ethnic separatism or self-determination, and the key leaders usually accepted autocracy as a necessary step towards winning the war. Overall, it's important to note that the white movement was rather disorganized, and several of its generals and leaders all had their own ideas and goals that they wanted to achieve out of the Russian Civil War.

As for why the Bolsheviks won, it ties in a bit with the flaws in the White Army mentioned first and foremost. Despite having a small opposition block within the party, for the most part they had a very centralized system of governance, and effectively consolidated their ideology among those who lived in their territories. They had one main goal and objective that they needed to achieve, that most of it's forces were working toward. A united and centralized army working towards one objective is going to be inevitably better then a loosely united and rather disorganized movement, that hits out in multiple directions at once. The second reason is overall, they had the homefield advantage in a sense. They had the majority support of the peasant classes of Russia, who were around 80% of the population at the time, had control over most of the industrial areas, which was huge in terms of things such as weapons production, and in general had more people to conscript and form an army with. Even though the whites did have foreign aid from a lot of Capitalist countries, for realpolitik reasons most of them couldn't commit more then ~20,000 troops and material aid at best, which doesn't help the most against the monolith that the red army was at the time.

In context to the song, what its about is mostly in the description, but the "little apple" and the "madam" comments are essentially poking fun at Kolchak, as he was their enemy.

Hope this helps!