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. Or just a story of things happening without incident.
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 for here, just trust us. You need it.
Conflict can 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 The Hero's Journey).
- Man vs. Man: The problem is another character (Bob needs to win over more votes than Alice does to become Class President).
- Man vs. Self: The problem lies inside the protagonist (Bob doesn't know how to express his emotions to Alice).
- Man vs. Nature: The problem comes from non-sapient sources (Bob's town is destroyed by a volcano, or Alice is sick).
- Man vs. Society: The problem is the social environment (Bob struggles to maintain his dignity in an ignorant community after receiving an Abomination Accusation Attack).
- Man vs. Fate/God: The problem is an undesirable destiny, which may also involve divine will (Bob does not want to fulfill a prophecy that he will lose his family).
- Man Caught in the Middle: Of other characters or conflicts.
- Male and Female: Quiller-Couch was persuaded to replace 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.
- Man vs. the Unknown has also been proposed as the type that codifies Horror, where the enemy is the incomprehensible, otherworldly or extraterrestrial.
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, a Zombie Apocalypse would thus be Man Vs. Nature. Going by this model, Man vs. Society and Man vs. God/Fate would both become Man vs. Nature, and Male and Female would become Man vs. Man (Man in the Middle and Man vs. Machine, on the other hand, could be either one depending on the plot).
Some stories may seem like they don't have a conflict, but if they have factors within the story that still present a challenge, then their story still contains the third type of conflict. No Antagonist describes this kind of conflict without an antagonistic force.
However, it is possible to have stories that actually don't have a conflict; such works tend to be non-fiction or experimental/postmodernist stories. Some episodes of Slice of Life series can also qualify, especially those following the Kishōtenketsu formula. Works without stories can also be considered as lacking a conflict (video games may even have a conflict but no story, letting the players create their own stories through the conflict). Such works are ultimately exceptions and not the rule, and in general, if it's a conventional and engaging story, conflict drives it.
It could be said that self-insert stories are weak because there is no conflict in how the self-insert achieves things, or that the conflict is so weak and ineffectual that it still comes across that way (as with poorly-written stories starring an Invincible Hero). Conversely, some works come off as weak because the conflict is too grave and/or appears to have no palatable solution. Like many other things, it's wise to strike a balance between the two.