A first-person point of view in a short story gives the reader a direct glimpse inside a character with no other intermediaries. The narrator is the "I" of the story and conveys the story through his subjective lens, as it were. Speaking in the first person is advantageous in a short story in that it lends an aura of authority. Nonetheless, to develop this point of view well, you should remember that certain limitations attend a given perspective. If you are the narrator speaking exclusively to your own experience, you will not be able to give insight into what other people are experiencing. To be effective, the point of view must reflect how a single-referencing view operates in real life: often, other people or situations must point out what a person would otherwise not know.
Like the first-person point of view, the point of view of the central character is a single point of view that looks inside and indeed, fuses with the perspective of one character, namely the protagonist. There is no intervening narrator to tell us what is happening outside of the perspective of this character. We are not made aware of other points of view that could distract attention from the thoughts and perceptions of just this one person. In developing this point of view, the key is to be consistent. Sometimes there is the temptation to bend the rules of point of view to learn more about the central character than she could actually know herself, but in doing this you can weaken the story.
In a short story, an author might switch back and forth between two different points of view, revealing what is going on inside several characters' minds. However, for the purposes of a short story, where brevity and word economy are paramount, this approach may not produce the desired effect of drawing the reader into identifying with a character if these points of view are fragmented. Stay with one point of view for the most impact.
An all-knowing narrator can often be found in short stories. An omniscient point of view, in addition to being able to jump between characters to reveal their respective thoughts and feelings in a seemingly objective way, is also able to convey to the reader additional background information that other points of view would not be privy to. Like the split point of view, this type of point of view requires strict handling in a short story. You often do not have the liberties available to you in a novel to engage the reader on multiple levels. Scattering your forces through jumping around or not making the point of view a unifying prism through which to perceive events in a story can disrupt the reader's engagement with its telling.