Does a story always have to have a definite conclusion, or can it be a mysterious ending, where it is up to the reader to decide what happened?

Generally, the conflict in a short story needs to have some sort of resolution. Without this closure, a story can feel unfinished and leave the reader frustrated. In that sense, yes, stories should have conclusions. However, resolutions in short stories usually aren't comprehensive. They tend to leave some lingering questions for the reader and this lends the story authenticity.

In Raymond Carver's "Cathedral," the narrator dreads the arrival of his wife's friend, a blind man named Robert who is coming to spend the night. The narrator is uncomfortable with his visit, and the fact that he's blind. At the end of the story, a cathedral comes on television and the narrator attempts to define it for Robert. He ends up drawing one, Robert's hand resting on his so Robert can experience the shape of it. Partway though, the narrator closes his own eyes and continues to draw. Robert asks him how it looks and without opening his eyes, he says: "It's really something." This is a small moment, but it's one where the narrator's perception changes. And this is all the conclusion this story needs. Notice the questions that still linger: Will the narrator become more sympathetic or accepting of those unlike himself? Will he form an actual friendship with Robert? Will he be relieved when Robert leaves? We don't know. But we do know this: the narrator was able to see outside his own, limited experience of life.

Think of an ending as a step in a particular direction, as opposed to an all out commitment to that direction. In Flannery O'Connor's "Everything That Rises Must Converge," Julian is disdainful toward his mother and her pleasures, such as her new green and purple hat, and the reducing class she takes at the Y to lose weight. At the end, when his mother has an attack after they get off the bus, Julian's true need for her comes out. We don't know how Julian will behave from then on, but his regret in that moment is palpable and revealing.

Even the most open endings offer some sort of resolution. Percival Everett's "The Fix" opens with Sherman being beaten up because he refused to fix what others wanted him to. A shop owner takes him in and he becomes a part of this community. We learn Sherman can fix unexpected things, like foot massagers earlier deemed unfixable, and the squeak in a shoe. And his fixing extends to people: He "solve[s] the Morado woman's sexual identity problem" and revives a woman who died in an accident. The people in the community become desperate and demanding of his ability. He's chased to a bridge with a "long drop, which no one could hope to survive." The mob presses in from either side. The story ends this way:

Sherman stepped over the railing and stood on the brink, the toes of his shoes pushed well over the edge.

"Don't!" they all screamed. "Fix us! Fix us!"

Does Sherman jump? Or does he climb down and face the crowd? Does he survive? The story doesn’t say. But this ending resolves an important element of the conflict. Sherman will continue to suffer at the hands of others because of his ability. This may not be an ability he will be able to use without immediate danger to himself.

The degree to which your ending is enshrouded in mystery can vary, but you do need to offer some resolution. And be careful to avoid endings that are so vague the reader doesn’t have any indication of what to make of it. A definitive resolution is part of what makes a story an actual story, as opposed to an anecdote, or a mere collection of actions strung together.