I think Trotsky once said something along the lines of "if the reactionaries had won the war, the word for fascism would have been a Russian one."
I suppose that's possible. Who knows what kind of governmental system would have been established in Russia, considering how fragmented and decentralized the whites were? Its certainly possible Kolchak, or maybe even Wrangel emerges as a strongman dictator of a rightist Russia. Since they've just won a civil war against the bolsheviks, the resultant system would probably be strongly authoritarian and violently anti-communist, so something like an ATL equivalent of fascism. I also expect it to be less than friendly towards the Jews, considering both traditional Russian anti-semitism and the prominence of Jews among the ranks of the bolsheviks, something that white propaganda drew much attention to IOTL. I doubt anything on the scale of the Holocaust is coming, but you might say a reinstitution of the more stringent Tsarist anti-semitic laws. So Russia itself might actually be the best bet for an ATL version of 'fascism' in a world where the reds are defeated.
As for the fate of communism in the world as a whole, I think its prospects look considerably worse than IOTL. The fact that there was one country, and a powerful one at that, actually ruled by a socialist party and operating according to socialist principles, was a major boon to the international communist movement IOTL. Most communists and many labor and workers' movements across the globe looked to Soviet Russia as an inspiration, and as proof that a socialist society was possible. They don't have that here. So much of the wind is taken out of the sails of the communist movement. There will certainly be no Cold War--at least not one anything like ours. No Nazi Germany, and maybe no Fascist Italy. No Spanish Civil War. I also expect you might see a less restrained capitalism, since the specter of bolshevism won't be hovering in the background to spook capitalists and capitalist governments into concessions to their workers.