Voice is the unique way we use and put words together to communicate. If two people were delivering the same bad news—that an employee's mother is in the hospital with pneumonia, for example—each would communicate that differently. One might say, “You should know, you're mother was taken to the hospital earlier this morning." Another might say, “Honey, there's no easy way to do this so I'll just say it: your mom's laid up in the hospital." Voice is much like a fingerprint; each person has one that is distinctive and identifying.
In fiction, voice comes through each character's dialogue and, when using first person, the narration itself is in a character's voice. (In third person point of view, it's the voice of the narrator, someone who is not a character in the story.) There's no single way to create voice. Instead, it is a result of all the tiny stylistic choices, like sentence structure and word choice, that you make along the way.
For instance, in Raymond Carver's “Cathedral," the narrator is waiting for his wife to return home from picking up a friend she hasn't seen in years, a blind man named Robert. He thinks about his wife's past and her relationship with the blind man:
Anyway, this man who'd first enjoyed her favors, the officer-to-be, he'd been her childhood sweetheart. So okay. I'm saying that at the end of the summer she let the blind man run his hands over her face, said good-bye to him, married her childhood etc., who was now a commissioned officer, and she moved away from Seattle. But they'd kept in touch, she and the blind man. She made the first contact after a year or so. She called him up one night from an Air Force base in Alabama. She wanted to talk.
The voice is casual, which is apparent in the phrases “So okay" and “I'm saying," and the use of “etc." The sentence structure also betrays the casual nature of the voice: “But they'd kept in touch, she and the blind man." First, he uses “they" and then clarifies exactly who he's referring to at the end of the sentence, something we often do in conversation when we're thinking on our feet.
Contrast this with the narrator in Ethan Canin's “The Palace Thief," who is a history teacher at a prestigious boy's school:
I tell this story not for my own honor, for there is little of that here, and not as a warning, for a man of my calling learns quickly that all warnings are in vain. Nor do I tell it in apology for St. Benedict's School, for St. Benedict's School needs no apologies. I tell it only to record certain foretellable incidents in the life of a well-known man, in the event that the brief candle of his days may sometime come under the scrutiny of another student of history. That is all. This is a story without surprises.
Word choices like “foretellable" give this character's voice more polish, and the deliberate and complete sentences convey formality. We could “listen" to both characters—from “Cathedral" and “The Palace Thief"—and know, without being told, who is speaking. And that is precisely what makes a voice distinctive.