In the first game you could join one of three factions - Future War Cult (FWC), New Monarchy(NM) and Dead Orbit (DO). Lore wise each faction came about for varying reasons which off the top of my head I can't remember. In Destiny 1 , you join a faction and you gain xp for it and each time you level up you get given some armour or weapons. There used to be differences in the armour they gave as to what stats got boosted, for example FWC used to speed up super and grenade recharge at the cost of slowing down your melee charge. This got removed later on in D1 and each faction offered varying stats on their armour which, to me at least, it then became irrelevant which faction you were part of (short of it you were trying for a specific weapon like Hung Jury).
I was FWC early in D1 purely because to me, super and grenades are more useful to me than a melee. As above though, once this got removed though, I stopped caring about factions short of trying to get some of their specific weapons. I hope they have brought back some differences in the factions so I'll wait to see if/what they are before aligning myself with one. If there is no, or very little difference, then I'm leaning towards DO or NM as their shaders were cool in D1 and FWC always looked shit.