In order to write good dialogue, you have to be able to hear it. This takes a lot of listening, which of course you should be doing all the time.
But here's a way to practice listening, and then to develop it in stages into writing.
Pick out some movies or TV shows you like, and watch them in some format (such as DVD) that you can control. Pick an interesting character and listen to that character's speech over and over. And here's the crucial step: Work on imitating that character's style of talking. Start by just repeating the character's actual dialogue. Eventually throw in ad-lib dialogue in that character's style. Do it repeatedly until you can do an impersonation of that character. You want to be able to talk like that person, and if you can, then you know you are hearing that voice.
When you can hear a voice inside your head in that way, you can write in that unique voice.
So, once you have internalized a character's mode of speaking, sit down and write a short speech that you think that character might say to a friend about something going on in his or her life. At this point, you can begin to think about what kind of word choices, phrasings, hesitations, sentence lengths, interjections, and so on, are used by that person. These are all the verbal characteristics that make that person's voice unique.
Do this for enough characters, and you'll start to learn what it means to hear a character's voice inside your head, and you'll start to find yourself creating new voices to replace the one you felt stuck in.
And by the way, screenwriters often do something pretty much like this in order to get their dialogue the way they want it; basically they play the character by speaking the dialogue out loud to see if it works.