It mostly depends on the game. In D&D, I tend to have one over-arcing campaign plot, with PCs having subplots that tie into the main plot. Not all PCs have these, though - some PCs don't want to bother with having subplots of their own, I've noticed, so I don't try and force these things on people who aren't interested. Other characters develop into a subplot on their own, in the course of the game. Most adventures will have something to do with this main plot, with very few that are purely extraneous to the central element.
For other games, I tend to be more episodic, with elements occasionally tying into a main plot. This is how I did it in Deadlands or my historical fantasy game. They'd occasionally get pieces of the larger puzzle from a side element of whatever their main focus was on, and as the game progressed, they started to get suspicions of what was going on. Respectively, those plots were "Somebody's trying to open a portal to turn Deadwood into a Deadland", and "The British military is run by demonologists - now what?"