I have been with potential buyers recently who are eligible for the tax credit and have asked this question. The first question is understandable but the second question I was concerned about. They were under the impression that since it is 2010 they had to file for the tax credit on their 2010 taxes (which makes since). This is not the case at all. If you are eligible for the tax credit, and you close on a new home before June 30, 2010 you can file for this tax credit on your 2009 taxes.
I am not an accountant, so I will not go into to much detail about the actual process of the filing, but if you have already filed your 2009 taxes, you can amend them with the tax credit form. The form you are going to be filing is the 5405 First Time Homebuyers Form (http://www.irs.gov/pub/irs-pdf/f5405.pdf). If you have not yet filed your taxes you must attach this form with your filing.
Once you have filed it is only a matter of time until you will receive your tax credit. I am not going to estimate how long it will take the IRS to drop your check in the mail because that would be foolish. There is a great Q & A site on the IRS site that can answer a lot of other questions you might have about this credit. You can get to it by going to: http://www.irs.gov/newsroom/article/0,,id=204671,00.html.
I hope this is helpful.