Go to a public place such as a park and really listen to the way people speak to each other. Bring a notebook and pen with you and try to discreetly transcribe bits of conversation you hear. Study your writing. If you're writing word for word what the people around you are saying, you'll notice lots of slang, incomplete sentences, and contractions.
Write new dialogue or revise existing conversations you've written, keeping what you've learned from your observations in mind. Always use contractions unless your character is especially formal. Use "um..." or "er..." if you want to show that your character is thinking about what he wants to say. Allow characters to break off mid-sentence and use sentence fragments.
Watch how your characters take turns between speaking. In real life, people interrupt each other or try to finish each others' sentences. Work this transition into your back-and-forth dialogue.
Consider each of your characters' ages, cultural backgrounds and personalities, and ask yourself whether you can change how each character speaks to reflect who they are. For example, an American teenager from New York City will speak differently than a seventy-year-old British woman. Depending on their personalities, some of your characters might talk more than others. Word choice also changes. While the teenager might use "like" a lot, the British woman would likely use the word "nappy" instead of "diaper."
Make sure that you punctuate and lay out your dialogue properly so that readers can follow it. Every time the speaker changes from one person to another that speaker should get a new paragraph. Use commas to separate conversation from dialogue tags, such as "Melissa said" or "she explained." For example, you would write: "I don't enjoy peas," Melissa said. Note that you don't need a comma if your dialogue ends in an exclamation mark or quotation mark. For example, "I hate peas!" Melissa exclaimed.