This is the basic problem to overcome in a story, the driving force. If you don't have Conflict, you don't have a story.
More than any other trope, save for the Characters who are in a conflict, this is vital to fiction. You can likely find loads of theories and essays on why this is so, but here just trust us. You need it.
Of course not every work in media needs conflict, but those tend to be non-fiction; some types of Slice of Life series (specially those of the Iyashikei genre) can also qualify. If it's a story or game, conflict drives it.
Conflict can also come in many forms. According to Arthur Quiller-Couch, there are seven kinds of conflict, creating seven basic plots (Not to be confused with The Seven Basic Plots by Christopher Booker, which articulates a theory closer to that of Hero's Journey).
- Man vs. Man, as in Character(s) vs Character(s). This could be the Big Bad, a monster, a robot, a friend, aliens, or anyone, as long as it's another character, major or minor. After all, The Villain Makes the Plot.
- Man vs. Self.
- Man vs. Nature or other Hostile Environment. or a Situation of some sort. Even Death Traps could count here.
- Man vs. Society.
- Man vs. God/Fate, or anything cosmic like those.
- Man caught in the Middle, of other characters/conflicts.
- Male and Female. Quiller-Couch was persuaded to remove the "versus".
Going beyond Quiller-Couch's list, there is also
- Man vs. Machine, as in machinery. Most commonly told from the perspective of a worker being replaced by a machine.
Traditionalists boil it down to the first three, redefining "Man" as a defeatable entity and "Nature" as anything that has to be survived or changed rather than defeated. According to the three basic conflicts, Zombie Apocalypse would thus be Man Vs. Nature.
Now it seems that some fiction doesn't have conflict, but even then it's presented as a challenge, which is the third type of conflict. See No Antagonist.
It could be said that Mary Sue stories are weak because there is no conflict in how the Sue achieves things, or that the conflict is so weak and ineffectual that it still comes across that way (as with Anti Sues and Boring Invincible Heroes).
- Contrived Stupidity Tropes (when the conflict comes from contrivances)
- Internal Conflict Tropes
- Rule of Drama (other Rule of Index tropes are for different reasons)
- Violence Tropes