How the faction choice works
All players will soon be divided into two hostile factions. Talk to Mikhalych and Valera after completing "The Protective Dome" quest to join one of the factions. You may as well refrain from it and keep playing by yourself, without belonging to any camp. Think twice before you make your choice! There's no way to change it later.
(Winner takes whole World)
Entering a faction
Talk to an NPC (Mikhalych or Valera) and accept their terms if you want to join a faction. If you choose to join the Black Sunset, you won't be able to communicate to vendors, storekeepers and other NPCs of Solnechny. If you're coming to the Confederation side, the NPCs of the Black Sunset Camp will stop talking to you.
Don't forget to take your things from the opposing camp before taking sides. That is, if you're leaning toward the Black Sunset, collect your belongings from Solnechny, and vice versa, grab your staff from the Black Sunset Camp if you're joining the Confederation. You won't be permitted to enter the hostile territory from now on.
Faction rejection
You should explicitly say no to Mikhalych or Valera if you wish to remain factionless. Be prepared to face harsh conditions and difficult trials, though. You have to show the real will to live and go through an incredibly difficult path to come back from the far-off wasteland.
It should be noted that an explicit faction refusal doesn't necessarily mean that you're taking the opposing party to the conflict.
Neutral path
So, here's your third option: you may decline both Mikhalych and Valera's invitations. If you opt for it, you won't have access to faction quests, your communication with faction NPCs will be limited, but it won't have any impact on your relations with neutral NPCs, such as Gosha or Kuzmich. However, you'll be free to visit both faction camps, as long as other players allow you.
Once a decision is taken, you can't change it. Consider your actions carefully!