First, from the point of view of the characters in the game (not the people playing), there are two basic ways to start an adventure:
1) Hope: The PCs want something to happen and go after it.
2) Fear: The PCs want something to not happen (or stop happening) and take steps to prevent it.
Some things about this:
-Lurking behind these is the idea that the players do or do not want something to happen (they want to level up, they do not want to suddenly not get to play with their current character 'cause it just died, etc.)
-Villains, rogues, and explorers are traditionally motivated to go on an adventure by the hope of making things better for themselves whereas heroes are more traditionally motivated by fear of the bad guy doing something bad. There are lots of exceptions and different ways to read this but if you click on that link there you'll probably see they're all covered so if you want to argue about whether Frodo was more motivated by hope for a better world or fear of Sauron being the boss of all the stuff and shit, then just remember, as an adventure Frodo's story starts with him trying to prevent something. Anyway this isn't that important to the point I'm making here now...
-Either way, all these Ways To Start An Adventure are highly dependent on 2 things: what your individual PCs (and players) want or don't want, and whether you're starting an adventure ex nihilo or doing it in the middle of a campaign.
-Hope is often more interesting than fear, theoretically, because it gives the players more options, however fear is often easier and more efficient for squeezing the PCs out of indolence and into action. Plus with hope you need to know what your players want, whereas with fear you don't.
-I am going to limit these Ways to Start an adventure to traditional RPG systems where player input into the setting, events, world, etc. is limited to what their own PC does or is otherwise limited by GM fiat. Otherwise the task is actually a lot easier since asking the players what adventure they want to go on is sort of programmed into the system one way or another. "You want to do a murder mystery?" "Yes""Ok, you're looking at a body..."
Ok, ways to start an adventure:
-Wake up incarcerated Love this one. All the GM needs is an interesting location surrounding the PCs and the first day of adventure writes itself. Downside: Though as a campaign starter it rules (hey, every campaign has to start somewhere) you usually can't use this in the middle of a campaign unless you're a total dick. Oh yeah, I know you checked into an inn at the end of last week but now you're in jail, sorry about that. I elided time, GM's prerogative and all...
-Geographical isolation The PCs are in some location surrounded entirely by adventure locations (a typical sandbox set-up). This is essentially the exact same as "Wake up incarcerated" only on a bigger scale and with hope instead of fear as a motivator. There are more options off the bat, usually. The PC chooses someplace s/he wants to go rather than knowing s/he has to get out by any means possible. Downside: Unless you want to give the PCs basically only one place nearby to go (in which case, why not just start the adventure there?) then you have to have either a lot of locations prepared or be ready to improvise a lot.
-Geographical isolation plus "stick" This is like Geographical Isolation only with one or more "sticks" (as in "carrots and sticks"): you're out of fuel, you're out of food, you need spare parts. Important: the PCs don't necessarily know where the food or fuel or spare parts are so it's still an exploration game. In reality, even ordinary Geographical Isolation has the stick of boredom for the players if they don't pick an objective--but what objectives they pick and when they'll stop is a little more clear if there's an in-game stick.
-Roll Initiative! Like Wake Up Incarcerated (WUI is better 'cause you start with some player interaction and problem solving) but it's more flexible since you can use it in the middle of a campaign. Downside: If the PCs win the fight and aren't captured, it won't automatically start an adventure--will the PCs wonder why they were attacked? Not necessarily. Basically you have to think up another hook/starter on top of this to be sure you start an adventure with it.
-Hey, Look--Shiny! Pure Hope here. The PCs see something they want remotely, behind some barrier, hear about it from an NPC, notice a very obvious sign of it in the environment or in the hands of another. Like always with hope, this only actually starts an adventure if you are sure the PCs will want the thing seen. A common version of this is any time the PCs are set up to notice that some NPC will pay money for some thing being done. Most classic-era published D&D adventures start this way.
-Curiousity Something strange happens. The PCs look into it and find adventure. Downside: Unless the strangeness affects them in a bad way (in which case it's basically Valuable Thing You Had Is Taken--see below) they hardly ever do look into it, since they got their own shit to do, unless there's an element of...
-Genre Obedience The players (not PCs) know that a hook is being placed in front of them and agree to go along with it because they know this is where the adventure that the GM bought or thought up is and know that if they're agreeing to roll up characters and play at all then they're agreeing implicitly to do this thing because they thought it would be fun and if they don't they have no-one to blame but themselves. Your uncle dies raving about a mysterious cult. So what? He was obviously crazy...No, wait, this is Call of Cthulhu--if you don't decide to investigate you might as well not play. Downside: While not necessarily the horrific burden some people make it out to be when discussing it in the abstract, this way to start adventures gets less and less useful as a campaign progresses and the PCs develop distinct personalities and the players develop interests in- and aversions to- specific parts of the setting.
-Valuable Thing You Had Is Taken On the one hand: this sure does work. A player who would not cross the street for a mountain of Treasure Type M will fucking scour the solar system to get back a pair of tweezers an NPC stole. Downside: if done improperly, or if done with some item or ability the player strongly identifies with the character (Spider-Man's web-shooters, say), some players will just get depressed and feel like they're not "getting anywhere" (even if the request to get the web shooters back nets them a lot of xp). They signed up to play a guy with web shooters and they want to play a web shooter guy, dammit! Or they found a magic ring and they want their ring, dammit! Plus it's hard to guarantee an NPC or monster gets an item away from a PC and if you build a whole adventure around that it could get mutilated fast if the PCs are clever enough to keep the macguffin. Make your thievery attempt at the end of a session. If it succeeds, then write your adventure around it.
-Threat To Something The PC Theoretically Values But Doesn't Own Someone tells the PC their country/religion/mom etc. is under threat or they discover evidence of it. This works fine so long as every member of the party actually values that thing or can be trusted to go along with the PC/player who decides to take action.
-Mission PCs get a choice: do a thing or have a fight with the people who want them to do the thing (if there are no consequences and they're just doing it for money or out of kindness, it's one of these other starters. If the PCs won't really have a fight and will just be annihilated if they don't wanna do it that's pretty much just a railroad and fuck that.) Pretty classic and easy all around. Downside: Like Genre Obedience, it can get old after the campaign's been going a while.
-PC Out Ahead of You Kind of like Hey, Look Shiny except instead of the PC finding out about something s/he wants, the PC decides to seek some objective that is implied by the setting but which the GM hadn't really given much thought to, like: Well, yeah, I guess you can sail east off the edge of the map, don't see why not...Downside: The major downside other than the fact that, for obvious reasons, you can't, as a GM, plan this, is that the other PCs have to stay equally invested or else you're dealing with a split-force campaign which can be logistically annoying.
-Parachute In And Roll Basically the GM has a whole interlocking miniworld ready to go, much of which is explained to the players up front. The PCs then get to make lots of decisions like where they start, on whose side they start, etc. etc. Downside: Requires a lot of work for the GM even before the game starts, and a lot of paying attention to this campaign material from the players.
-Looming But Not-Yet-Present Threat To PCs Themselves A lot like Geographical Isolation Plus "Stick", though the PCs do not necessarily have to be geographically isolated, they could just be in a well-defined world (or one the GM's willing to improvise up as the game progresses). Like Parachute In, the PCs will have to be given a lot of information in order for their freedom to deal with the threat as they please to be meaningful. If they are only presented with pre-canned methods of dealing with the threat, it's just a "Mission", really.
-The "Seinfeld" Like Geographical Isolation except the GM does not use geographical boundaries to slow PC progress. Like if they're suddenly all "let's go to India" then, no problem, they're suddenly in India later in that same session. The GM may either slowly introduce hooks and/or obstacles, like in Geographical Isolation, or bring in any other adventure-starter on this list. This can be entirely improvised, possibly based on what the PCs say while they're trying to figure out what's going on.
-It's A Long Story All the PCs are wherever they are for different reasons based on their history or personality or job. Like: you're the translator, you're the bodyguard, you want the Flower of Almon-Grealle, you need to kill three of the other PCs to get a reward, etc. The defining characteristic here is that each PC has a different motivation and they have to sort of argue it out in order to figure out even what kind of adventure they're having.
-Stages of Defined Choices The group has to make some decisions, right off the bat: though these are framed as in-character decisions specific to the kind of adventure the GM is going to run (which route will you take, what are you driving, etc.), they do not take place at the normal game time-scale and essentially help the players define the adventure. It's an in-between step: ike after the players have generated their characters individually, they have to work together to generate something else before they play. The nice thing here is it immediately gets the PCs invested and involved in the set-up in a cooperative way. Some other kind of way to start an adventure usually will have to be bolted on the end after this one.
-Combo Roll twice on this table..
That's all I can think of for now. I thought of about a million others that, after analysis, basically fit one of these options. If you think you've got one I totally missed, comment.