Digital signatures are used to establish the authenticity of electronic messages and documents. They are usually based on asymmetric cryptographic algorithms, such as the RSA algorithm. The validity of digital signatures is regulated by legislation in many countries, such as the Signaturgesetz (Signature Act) in Germany. Digital signatures are sometimes called “Electronic Signatures”.
By far the most common digital signature algorithm is RSA (named after the inventors Rivest, Shamir and Adelman in 1978). The RSA algorithm has been standardized (ISO, ANSI, IETF etc.) and been extensively analyzed by the cryptographic research community and you can say with confidence that it has withstood the test of time, i.e. no one has been able to find an efficient way of cracking the RSA algorithm. Another more recent algorithm is ECDSA (Elliptic Curve Digital Signature Algorithm), which is likely to become popular over time.
Digital signatures are used everywhere even when we are not actually aware, example uses include e.g. retail payment systems like MasterCard/Visa chip and pin, high-value interbank payment systems (CHAPS, BACS, SWIFT etc), e-passports and e-ID cards, logging on to SSL-enabled websites or connecting with corporate VPNs.