*UPDATE 2/14/10** Much of my previous information on freelancer taxes, business tax deductions, and estimated payments are quickly becoming outdated, or may no longer apply to many freelancers in the animation/vfx industry. See this post for more information: http://kylemohr.blogspot.com/2010/02/freelancers-guide-goodbye-1099.html
Estimated tax payments were almost a complete mystery to me, until yesterday when I was completing my return on Turbo Tax Online, and their handy help files had some info on it. There's no direct link to those help files, but I found a couple pages online from Turbo Tax's tax tips. They should clear up a lot of your questions (and mine!)
Estimated Taxes FAQ
Determining Estimated Tax Payments
If you'd rather have every possible info on the topic, which is not a bad idea just to be sure, check out what the IRS has to say about it here. That page seriously outlines the whole thing in detail, lucky there's a table of contents that you can sort through to see where the main info is at.
If you are brand new to this whole idea, estimated tax payments are generally required of you if you are freelance, because taxes are not taken out of every paycheck. This means that you estimate the amount of taxes you will owe for the current year, based on the past year's return, and pay quarterly estimated payments. If you don't do that, you get an underpaid taxes fine at the end of the year. That is about all I knew about it until Turbo Tax taught me more! Turbo Tax apparently even has a service dedicated to helping you make your estimated tax payments, which I will probably be using. Note: If you plan on using it to pay your taxes at the end of the year, you need the Home and Business version. It's more expensive, but has everything you need. If you use Turbo Tax Online, you don't have to pay for it until it is complete and you are ready to file and/or print your return. So far, the federal parts of the service seem more helpful than the New York state sections, but it still beats trying to figure out how to read the forms.
A lot of people have an accountant, which is actually not a bad idea, especially when you get to the point of buying a house, etc. But until then, tax software is probably the best bet to help you get them done.