Your first draft of a story may be messy, with a lot of unnecessary words and phrases. You'll probably find that as you edit your dialogue, it will become more succinct. Try to think in terms of speech patterns, and less about storytelling through your dialogue. Listen to how people talk and pay attention to what your characters are already familiar with. For your dialogue to be realistic, both the characters AND the reader need to believe it.
Keep Sentences Short
In general, keep sentences short. Oakley Hall, in The Art and Craft of Novel Writing, offers the rule, "One thought at a time and keep the lines short." Most people don't talk in perfectly formed, complex sentences. For example, in this passage from Raymond Carver's "What We Talk About When We Talk About Love," note how short and simple most of the sentences are:
"When I left, he drank rat poison," Terri said. She clasped her arms with her hands. "They took him to the hospital in Santa Fe. That's where we lived then, about ten miles out. They saved his life. But his gums went crazy from it. I mean they pulled away from his teeth. After that, his teeth stood out like fangs. My God," Terri said.
Cut Unnecessary Dialogue
Paring down your sentences may not be enough, however. Chances are, there'll be scenes you wrote for yourself, to get to the next part of the story. Cut any unnecessary dialogue. If it doesn't build character or advance your plot, edit it out. In the Gotham Writers' Workshop guide to writing fiction, Allison Amend explains it this way: "The realism of good dialogue is something of an illusion. Readers of fiction have a higher expectation for dialogue than the conversations of real life. Fictional dialogue needs to have more impact, focus, relevance, than ordinary conversation."
A conversation about the correct route to take when driving, for instance, is extraneous if it goes like this:
"So I think we should take Elm all the way to Lincoln," Mary said, the map spread across her lap.
"Is that really the best way?" Mel asked her. "What if we hit traffic?"
"But it's Sunday. We'll be fine."
There's no tension and nothing necessary is revealed here, so there's no reason to include this scene, though it is true to life. Presumably, these characters are on their way to something important: why not fast-forward to those key scenes, and leave out the logistics of getting there?
On the other hand, if the scene were to reveal something about Mel and Mary's relationship, something that mattered to the plot, we would keep it:
"Why aren't we taking Elm?" Mary asked.
"Did I ask your opinion?" Mel said, switching lanes a bit too quickly. "When you drive, you can pick the route. But I'm driving, so I'll pick the godd*@n route."
"Fine, fine," Mary said. With a sigh, she reached over to switch on the radio. "If you'd ever let me drive, then maybe I could," she said under her breath.
For more, check out How Do People Talk in Fiction? where you find information on how to edit your sentences so they sound like real dialogue, use dialogue "correctly", and learn more about when to use dialect. See also Writing Dialogue in Action Scenes. For the correct use of grammar when writing dialogue, read How to Punctuate Dialogue Correctly.