Technically, the story seems to be third person limited omniscient. It's told in the third-person, and only follows Della. We don't see what Jim is doing during the story, and once he does show up, he remains closed to us: we don't know what his reaction to Della's hair is any more than Della does.
We can't be entirely satisfied with this classification, though, because the narrator has such an independent personality and seems to know a lot more than Della does at times. He's "The Storyteller." It's as if he sees everything, but usually limits himself to Della's point of view by choice for storytelling purposes. If the narrator described everything that were going on, he'd ruin the surprise ending.
We know the narrator is really more like an omniscient being, though, because every so often he "zooms out" to make much more general pronouncements that fly way above the action of the story's characters. The most obvious of these is at the end, when he mentions "the magi" (to which Della and Jim are totally oblivious). But there are other places too, like when he zooms out from the weeping Della to describe the flat. There are also all those moments when he makes a more universal remark about "the way life is," such as, "Life is made up of sobs, sniffles, and smiles, with sniffles predominating" (2).