Of course everything depends on your particular situation, but basically, by claiming you as a dependent, your parents can deduct an amount from their taxable income each year. If you are no longer a dependent for them, they can no longer deduct that amount. You get to deduct the amount from your taxes. Your taxes should go down (again, depending on your circumstances), but your parent's taxes will go up. Since they are likely in a higher tax bracket than you, their taxes might go up a little more than yours go down.
If you have more specific questions, feel free to ask, I'm sure there's enough folks with tax info for ya!