I am 15 years old and I just started freelance development. Currently I am making $200-$400 a week and I live in the US. Anyway, my question is do I have to pay taxes and how?
Since you're a minor (I'm assuming you also live with your parents), you generally don't need to file your own tax return unless you earn quite a lot of money on your own.
How much is quite a lot? At least $6200 a year, for earned income only, which in your case it looks like you have. If you have un-earned income (interest/dividends/capital gains) then the limit is lower: $1000.
Since you are above the threshold, you'll need to file your own tax return (your parents will need to file it for you, more likely). The detailed instructions on what to do are published by the IRS in Publication 929.
As a sidenote, you may want to consider depositing your income to a Roth IRA account, where it will grow tax free and provide a very nice cushion (tax free as well) for your retirement in the future.