What (US) tax form do I need to fill out with a company as an independent contractor?

W-9.

BUT, before you do, got to www.irs.gov and get yourself an EIN - Employer Identification Number.

It will protect your identity from the possibility of identity theft. You don't want to give out your ss#.
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Judy is right that you don't *need* an EIN, but as a bookkeeper and tax preparer, I strongly suggest you get one. It seems every time you turn around, you see another story about ID theft. Once your identity is stolen, you cannot just change your social security number. It's yours for life. And so are all of the credit problems that go with ID theft. Why not take one simple step that could prevent a lifetime of hassle?
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the tax lady is wrong:

From the W-9 page 3 :
"If you are a sole proprietor and you have an EIN, you may enter EITHER (emphasis added) your SSN or EIN. However, the IRS prefers that you use your SSN."

http://www.irs.gov/pub/irs-pdf/iw9.pdf

So you do NOT have to give your SS#.

Also, your have to report EVERY dollar earned, whether it is cash received or barter exchange. Whether the income is taxable or not is what the 1040 and it's various schedules (C, E, etc) determine.

Give them a W-9 form. You can use your ss#, don't need an EIN.

Form W-9

The W-9 form explicitly states that you cannot use an EIN on it, you must give the payer your SSN unless you have incorporated or a partnership.

By the way, if you don't give them a W-9, they have to withhold 28% of every payment as backup withholding. And that starts at $1, not $600.

Lastly, if you are paid $400 you are required to report the income even if you don't get a 1099-Misc.