Situations:
1) Are the result of player initiative. The player defines the goals or chooses the foes.
2) The DM doesn't have an end state or victory condition in mind. The DM doesn't want to 'accomplish something'.
3) Generally involve a fairly small amount of preparation, and are often initially created on the fly with no preparation and additional preparation is done after that session. Dungeons are usually small, on the order of single lairs or side tracks, and the map is usually created on the fly or from some generic stock. Large potential dungeons are generally not initially treated as dungeons ('The Cathedral of Lado') and the layout is handwaved for now, or else are generated on the fly by some means.
Some examples, of situations...
"I want to rob this house."
"I want to go on a pilgrimage to my patron deities holy site in order to make a special sacrifice."
"I want to find a wizard who will be willing to teach me the fly spell."
"I want to go hunting in the forest south of town."
"I want to call on the daughter of the local Baron."
"I've decided to avenge myself on the Duke's seneschal, here is my plan..."
"I need to find a means of curing Bob's insanity."
"I've decided to settle down here. I want to recieve license to crenellate from the local Lord."
"My merchant business would be more profitable if I could break House Agruille's monopoly on silk importation - I want to go see the Shalasan ambassador who owes us a favor."
"I want to go to Griffin Vale to see if I can acquire a griffin egg, for the purposes of rearing a steed."
These aren't adventures in the normal sense. They represent portions of the character's life. They may cause conflicts, intruige, combat and all the sorts of things we associate with adventure, but they don't have a the sort of structure we normally associate with an adventure.
Adventures:
1) Are the result of DM initiative, usually some sort of scripted event (a 'hook') designed to either lure the PC's into cooperating ('a huge treasure is to be had', 'I'll pay 3,000 now, and another 12,000 when we get to Alderan', 'somone is murdering priests of Lado') or to force their hand ('an innocent is in distress', 'your leige commands', 'your sister has been kidnapped').
2) The DM has a specific set of events in mind and set out who is an ally, an foil, and a foe. The DM has a specific end state in mind when the adventure will be resolved (or not), such as 'The dungeon is explored', 'The treasure is recovered', 'The villain is thwarted', 'The innocents are rescued'.
3) Generally involve alot of DM preperation to pull off successfully, much of which is usually complete before the session begins.
Examples of adventures:
1) Virtually all published modules, and certainly any modern ones.
2) A dungeon to loot.
3) A series of scripted highly scipted events or preplanned encounters.
Some things really blur the line. Generally I'd classify a mega-dungeon as an adventure, but a well designed mega-dungeon can approximate a setting, to the degree that the PC's are the ones setting the particular goals and quests that they wish to accomplish. That represents something like sandbox play within an adventure. Likewise, the DM may introduce overall meta-plots to the setting such as, "Two empires are at war." or "There is a rebellion in the land." and this may provide reoccuring color and context even if the DM has no particular idea of how the players should act. The DM may have no idea which side the PC's will end up on, having chosen neither as the clear foe or villain. The PC's may ultimately decide for themselves which sides NPC's are more compelling, which side provides the best oppurtunity for them, which side has the more real grievance, or whatever. Or they may decide to just not get involved. This represents something like adventure play within a sandbox.
Really, alot of this is going to depend as much on the players as the DM. Alot of players just want to be taken for a fun ride. And even if the player's are happy playing in a sandbox, you may want to throw an occasional adventure at them just to see how they handle a curve ball, heighten the tension, or recapture flagging interest. On the other hand, if the players are showing signs of being tired of being on the railroad, or seem to be trying to get off, it may be time to just slow things down, put larger plots temporarily on hold, and let the player's fulfill ambitions of their own.
If it involves conflict, intrigue and combat in the context of a roleplaying game, I have a hard time coming up with a reason not to call it an adventure.
