Some storis do not name the narrator. A possible explanation for this is that the unnamed narrator becomes every human being, thereby enhancing the universality of the short story. In other words, the narrator represents anyone who has ever acted perversely or impulsively-and then had to pay for his deed.
Is there special "literally term " for that ??
It sounds like what you are asking about is an omniscient narrator. The omniscient narrator does not exactly become every character, but he or she is all knowing and can therefore see into the lives of each character, therefore giving the reader a more universal point of view.
Charlotte Perkins Gilman's story, "The Yellow Wallpaper" has an unnamed narrator who writes in the first person, so the point of view can be either third person or first person. But, as mentioned in the previous post, the anonymity of the narrator provides the reader the opportunity to relate to this narrator. In Gilman's story, for instance, the unnamed narrator could be any new mother of the Victorian Age who suffers from post-partum depression.
If the narrator has divulged information about deeds of perversion, or about any deeds committed by him/her at all, then we are dealing with a special kind of narrative perspective (if this narrator does not engage with other characters in the story proper). You might call this a "frame narration" which would provide context and subtext to the story proper. This isn't a generally used term though.
If this character is not named, yet describes things that he/she has done and also interacts with other characters in the story, then we are dealing with a nameless character and narrator and a first person narrative perspective.
We definitely need to determine whether or not the narrator is also a character before we can decide how best to describe this figure.
After reading your post, I would have to agree with the other posters. The narrator would be called a 3rd person omniscient or 3rd person limited. Although, I do have an issue with the explanation provided: "the narrator represents anyone who has ever acted perversely or impulsively-and then had to pay for his deed." I do not think that every narrator who represents the omniscient narrator is speaking from a point which only makes one think about a bad deed they have done. Instead, I think that the omniscient narrator, instead, provides the reader with unbiased point-of-view. Problem is, some omniscient narrators can be biased based upon the authors own tone towards the subject matter.
Outside of that, one can also find that "these types" of narrators can be unreliable as well.
If the unnamed narrator of the story is ALSO a character in the story then your comment would make sense and you would be looking at a first person narrator. In literature we sometimes refer to those kinds of characters as an "everyman character" because he or she represents a truth of humanity. Third person narrators are not characters and so they "float" above the story revealing a limited 3rd or an omniscient 3rd person perspective.
One term often used to describe such a narrator is "omniscient narrator" -- in other words, a narrator who somehow knows everything about all the characters and all the details of the plot, including the characters' inner thoughts and motivations. Usually such a narrator is never identified or given a name. In a sense, such a narrator has a "God's-eye" view of everything in the story.