A federal Employer Identification Number (EIN) is used to identify a business. Generally, all businesses need an EIN. Businesses with employees use their EINs for reporting federal income tax withholding, Social Security payments and unemployment insurance. Businesses can apply online with the IRS here.
As you may know, New Hampshire does not have a general sales tax or an income tax, but it does have some types of business taxes. To understand them and how they might apply to your business, you should start with this Overview of NH Taxes. Be sure to check out the Business Enterprise Tax and the Business Profits Tax links there. Check to see if your type of business is eligible for any of the other taxes listed.
State Unemployment Liability
If your business will employ people, you need to be aware of the unemployment taxes you will need to pay, and what happens if you lay off employees and how unemployment compensation works. Start with the resources available at the New Hampshire Department of Employment Security.