Jeez...I'm trying to think of the most common/popular that haven't been mentioned yet...
I've been in campaigns where all of the PCs are thrown together by happenstance. Not the "you're all from the same peaceful village" but simply "You're all in the same place at the same time and the shale hits the fan." so, at least initially, they stay together because they have a successful combat or two and figure staying together is their best chance of survival.
I've played in the "you're all seeking the same item/person/place (but don't necessarily all start out together) for your own reasons and join forces to increase your chances."
I've played the "you're all at an inn/tavern and show up to answer the same 'job posting' on the wall."
I have one where one of the PC's actually hires the other PC's. This only really works if you're willing to have one PC be wealthy enough to do this (or have a player who wants to be from/have some wealth/wealthy family of their own).
The way it worked was they were hired as protection from point A to B. Then, naturally, along the way they become embroiled in other plot devices/elements that arise so their reason for staying together evolves ...friendships form, romances maybe, sense of duty to thwart some evil scheme, pure survival, whatever it happens to be for each PC...beyond just the "getting paid."
Thus the adventure(s) becomes "what happens on the journey", not necessarily the initial perceived goal of "getting to point B."
I've done the "you all come from a land torn by strife" thing too...
- The Inhuman Invasion: the Goblin Wars, the Troll Wars, the Ogre Incursion, Pick-a-Humanoid-du-Jour and fight'em
- The Human Invasion: Empire X is stretching its influence and trying to take over your homeland...and the homelands of all of the other PCs (or maybe some PCs are already from conquered lands and looking for allies to push the Empire back?)
- Civil war: self-explanatory. Political factions, religious factions, members of the same royal family fighting each other for "their" throne.
- International war: Kingdom A v. Kingdom B, the Elf Kingdom v. the Human Kingdom, 2 dwarves and a clan of gnomes against the rest of the known world, etc.
- Rebellious uprising against an unjust ruler: This isn't a "full blown" civil war kinda thing, but a Robin Hood vs. Sheriff of Nottingham kinda thing...
- Refugee situation from some outside nation/realm
- Spreading disease/plague: Keep in mind, doesn't have to be a modern day "medical" type of plague...maybe a "madness" or catatonic condition is striking random people or magic malfunctions or crops are blighted and animals dying with no apparent "disease" cause or a "Creeping Curse" that simply brings serious, sometimes deadly, misfortune wherever it goes...
No matter what the source of the strife, you have to make sure that the conflict is serious enough/centered around something the PCs are going to care about a LOT to take up arms for one side or the other.
If the PCs have no investment/immersion/interest with Princess Prettyfingers OR Duke Hardball, then why should the PCs care/be involved in
their battle for supremacy? If none of the PCs (or their loved ones) have been effected by this disease spreading across the land, why are/should they going to quest for a cure?
Simply saying "this is what's going on in the world" is not usually, ime, enough to get players engrossed in the world. It
can be, with what would be (imho) "good" players. But not always.
Whatever the backdrop, I think the most important thing to keep in mind is the PCs have to (or at least
should) care about where they are/what's going on in the world around (and perhaps
to) them...and thus, why they are out for [adventure/loot/fame/revenge/et cetera ad infinitum] in that setting in the first place.
Have fun and happy "expositioning."

--Steel Dragons