First person, second person, or third person is, fundamentally, a creative choice on the part of the author. Simply put, “person” is a matter of point of view, expressed via pronouns.
In a first person narrative, the story is coming from exclusively the point of view (POV) of a present narrator—the main character in a story who is telling you what happened after it happened (past tense) or what is happening as it happens (present tense). Note the singular pronouns “me” and “I” in A Drink Before the War by Dennis Lehane:
She got me across the street and up the stairs, the queasiness steadily evaporating as the warmth of her skin and the feel of the blood rushing through her body reawakened my senses.
We sat down in the kitchen. I kicked Harold the Panda out of my chair, and Angie poured us each a glass of orange juice. She sniffed hers before she drank. “What’d you tell the Asshole?” I asked.
The trade-offs with first person is that we can establish a tightly focused experience for our readers, who we’re essentially compelling to “be” that one character, experiencing everything in the story (the other characters, the world, the plot, etc.) filtered exclusively through that one character’s perspective. The flipside being the loss of the experiences of all your other characters. We don’t know what anyone is doing out of sight of that one first person POV character. We can’t “cut to the villain” to see what nefarious schemes are being cooked up, etc. This can make suspense, which comes from an imbalance in information, more difficult to conjure, since our readers can’t know anything the main character doesn’t know, including some impending danger or difficulty.
Second person? Almost no one attempts it, and I think you’ll find few readers connect with it, since this is you telling your readers what they’ve done (past tense) or are doing (present tense) in the body of your character. Here the dominant pronoun is “you,” as in Diver’s Clothes Lie Empty by Vendela Vida:
The streets seem wild to you now—so many trucks and so much smog, and the potential for motorbikes to bump against vans. The buildings around you are ugly. They once were white but now are dusted with soot. There’s nothing to look at through the window except traffic. You can’t wait to check into your hotel room.
Honestly? I’ve seen it done well twice, by Iain Banks in A Song of Stone and by J.M. McDermott in Last Dragon. If you’re up for it… good luck!
More common in genre fiction is third person, in which we’re able to jump into the experiences of more than one character, describing what that character s experiencing in that moment. Pronouns are specific, “he,” or “she,” or characters are identified by name or other identifying nouns, as in the third person, present tense The Girl With All the Gifts by M.R. Carey:
When she thinks of it like that, the decision is made for her. She needs to start with the child who shows least impairment of all. The child who, despite having as high a concentration of fungal matter in her blood and tissue as any of the others, and more than most, somehow retains a genius-level IQ.
She needs to start with Melanie.
Third person has the benefit of being able to establish that imbalance of information. We can now inhabit the villain’s experience while traps are laid then go back to the hero’s experience knowing danger is a few steps away—steps the hero doesn’t know not to take.
Third person also gives you some greater ability to play with time and continuity, which Sianne Ngai described in “The Gimmick of the Novel of Ideas”:
Direct speech by characters involves privileging what narratologists call scene, in which story and discourse time coincide. This dramatic tempo contrasts with those at which the novel uniquely excels: summary (fictional events unfolding over years are briskly accounted for in a single paragraph or even sentence) and stretch (a story event taking up less than a second is recounted over several pages of text). Theater cannot do stretch without recourse to special effects like film, which has to rely in turn on special effects like slow motion. Film struggles with summary, resorting to devices like montage or peeling calendars. Summary does not come easily to theater either, which manages it through expository speeches by characters. In short, when the novel’s dominant temporality becomes the “real time” of scene, as opposed to psychological stretch or historical summary, the novel is no longer in its technical wheelhouse but that of another genre. Indeed, stretch and summary are the only temporal modes in which an innovation entirely unique to the novel has been able to develop. Free indirect discourse, in requiring the grammatical third person, cannot take place at moments of direct speech by characters. Nor can it take place in the direct speech by narrators which gives rise to the “pause,” in which discourse time is maximal and story time is null.
Juggling multiple POV characters can sometimes get out of hand, leaving your readers trying to “be” too many people, and switching from one to another too rapidly. The only rule I feel is unbreakable in writing fiction is: one scene, one POV. Once you’ve put your readers into a character’s experience, stay there—at least for the space of one scene, then indicate with a scene break or the start of a new chapter that that connection has been broken and establish a new POV character in the next scene. Though you’ll easily find older books that ignore this rule—Dune is a notable head-hopper—I doubt you’ll find a book by an experienced author published (in America, at least) in the past forty years or so, that doesn’t adhere to one scene, one POV.
So then, for the next story or novel you sit down to write, which is best?
In “Revising One Sentence,” Lydia Davis wrote:
Sometimes I write about myself in the third person and sometimes in the first. Thinking about it now, I realize what determines this: If it matters that I’m the one doing something, if I am truly the subject, then I write in the first person. If it does not matter who is doing it and I’m simply interested that a person is doing the thing, then I write in the third person—i.e., I’m using myself as a source of material and I’m more comfortable writing in the third person because then I (the writing I) don’t get in the way of the character that may evolve from this action.
In any case, remember what I said up front about this being a creative choice on the part of the author? Well, if you’re the author, make that creative choice!
—Philip Athans
More on POV:
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Fantasy Author's Handbook 
The most interesting book I’ve read in 2nd person is If on a Winter’s Night a Traveler but Italo Calvino. It opens with the narrator saying you just bought a book called If on a Winter’s Night a Traveler. It’s essentially a master class in different writing styles.