National Insurance number
The National Insurance number is used as a reference number in the Pay As You Earn system, and also by the self-employed. It is also used in applications for Individual Savings Accounts (ISAs), to check that an individual has opened only one ISA in a tax year.
Social security
Social security is "any government system that provides monetary assistance to people with an inadequate or no income." In the United States, this is usually called welfare or a social safety net, especially when talking about Canada and European countries.