Underwriting refers to the process that a large financial service provider (bank, insurer, investment house) uses to assess the eligibility of a customer to receive their products (equity capital, insurance, mortgage, or credit). The name derives from the Lloyd's of London insurance market. Financial bankers, who would accept some of the risk on a given venture (historically a sea voyage with associated risks of shipwreck) in exchange for a premium, would literally write their names under the risk information that was written on a Lloyd's slip created for this purpose.
Read more about Underwriting: Securities Underwriting, Risk, Exclusivity, and Reward, Bank Underwriting, Insurance Underwriting