A SSN is a social security number, which can be obtained by US citizens, resident aliens, and certain other non US persons who legally work in the US. It is issued by the Social Security Administration.
An ITIN is an Individual Taxpayer ID Number issued by the IRS to certain non US citizens who are not eligible for SSNs but require a taxpayer number for certain IRS purposes. An example might be the spouse of a foreign worker who is allowed to accompany the foreign worker in this country, but cannot be employed. The IRS web site has an excellent description of the
ITIN
Sole proprietorship business may use their owners' SSN for opening bank accounts and paying taxes. If they have employees, they often will also have an Employer ID number (EIN) from the IRS. But under CIP rules, you will collect their SSN.
Same on some sole-owner LLCs (Limited Liability Companies), and certain limited types of trusts.
All other businesses and entities must obtain an EIN from the IRS.
Now, aren't you glad you asked?