TINs are unique numbers used by the IRS to identify individual taxpayers. For most people, that's their Social Security numbers.
Individuals who are not U.S. Citizens, nationals or permanent residents and, hence, do not have Social Security numbers use IRS Form W-7 to ask the IRS to assign them a TIN, known in this case as an "ITIN" or "Individual Taxpayer Identification Number" (discussed further in IRS Publication 1915). The IRS also issues "ATINs" or "Adoption Taxpayer Identification Numbers," which prospective parents use as temporary numbers for children in the adoption process.
Taxpayer Identification Numbers for businesses and other organizations are known as "Employer Identification Numbers."