If it's a small contract and your first foray into business, you can just need a tax ID number. In the U.S. your Social Security number works. If your working as a contractor you need to fill out a W-9, and you will be responsible for paying your own taxes. You don't always need a business name, in the beginning your real name can just be the busines name.
This can be the biggest pitfall. Make 10K... spend 10K, and then you in a panic when the taxman comes around looking for 3.5K of it.
You want to draw up some sort of contractual agreement. Clearly define what is your responsibility and who owns any work created. It's also a good idea to read the SL TOS and address any concerns with it in the contract. For instance... LL makes no claims that your data is safe, and if something were to happen to your inventory and you get 3 months of work wiped out, that could have some RL business ramifications.
I also like to write up a nice descriptive proposal document, that defines what your are responsible for, what your deliverables will be, and a timeline with some milestones for the project.
You'll need an address, since most companies pay by check, but there is no reason you can't use your home address. You also probably should expect to give out your real name and telephone number as well.
The suggestions above are for starting out and testing the waters. As you generate revenue and land bigger deals you'll want to incorporate, get a company credit card, open a bank account, and perhaps a new mailing address. A website is always good too

If you have more specific or private questions feel free to email me (Satchmo.Prototype@gmail.com). I've worked with a couple of RL business' in SL and our oppurtunites appear to be expanding.
Some of this stuff is subject to change, depending on what kind of business you are creating too. I have RL experience working as a private consultant, and have expanded that into SL consulting.