Short answers, pulled from the story.
A digital signature scheme consists of three algorithms: key generation, signing, and verification. Key generation selects a private key at random to produce a corresponding public key. Signing takes a message and private key as inputs to produce a signature, while verification accepts or rejects the message's authenticity based on the public key and signature.
Whitfield Diffie and Martin Hellman first described the notion of a digital signature scheme in 1976. They conjectured that such schemes existed based on functions that are trapdoor one-way permutations before Ronald Rivest, Adi Shamir, and Len Adleman invented the RSA algorithm shortly afterwards.
Shafi Goldwasser, Silvio Micali, and Ronald Rivest became the first to rigorously define the security requirements of digital signature schemes in 1988. They presented the GMR signature scheme, which could be proved to prevent even an existential forgery against a chosen message attack.
Storing a private key on a tamper-resistant smart card provides two-factor authentication because the user must enter a personal identification number or PIN code to activate the card. This ensures the private key never leaves the smart card and requires both physical possession of the card and knowledge of the PIN to generate a digital signature.
The first statute authorizing digital signatures appears to have been enacted in Utah in the United States. Other countries have also passed statutes or issued regulations in this area, including the 1999 EU digital signature directive and 2014 EU follow-on legislation that legally bind signers to document terms.