If you are not in business, you already have a social security number, you don't need any other tax ID number. If you are not allowed to work in the US (not a citizen, no green card) you can still pay taxes, the IRS just wants their money and doesn't enforce immigration laws. You would apply for an ITIN (international tax ID number) when you file your first tax return.
If you are in business, you only need an EIN (employer ID number) if you pay other employees. But, you can obtain an EIN and use it even if you aren't required to have one, it allows you to give the EIN to banks and contractors instead of your personal SSN. It takes about 20 minutes to get an EIN online from the IRS.
The tax requirements for a small business/sole proprietor/independent contractor are the same whether you have a tax number or use your SSN for business. @VolvoGirl outlined them above.
