An electronic signature is an electronic record of an agreement.

Contracts have been used for a long time to show that two parties agree on something. Often these parties will then write a document that they both sign to show this agreement. In the times of the internet, many of these documents are transmitted in digital form, but showing agreement is still needed. That is where an electronic signature comes in. 

The concept of electornic sgnature itself is not new. Common law jurisdictions have recognized telegraph signatures as far back as the mid-19th century, and faxed signatures since the 1980s. 

Electronic signatures exist in different forms. All forms can show that somebody agreed with something.  Some forms can also protect the data the person agreed with from being changed easily, or they can legally identify the person that agreed. To do this, ideas from public-key cryptography are used: digital signatures, certificates, and hash codes. An electronic signature often includes a timestamp to show when the signature was made. Much like cryptography, electronic signatures can be used for any kind of data, there is no requirement that the data signed has a specific format. 

Even though cryptography is often used, the term electronic signature has a legal meaning. This is different from the technical term digital signature used in cryptography. Many countries have made regulations so that  some electronic signatures   are equivalent to a handwritten signature for many purposes. 

There are different ways in which an electronic signature can be done. Many countries have  standards as to what such a signature must look like. Examples for such regulations are eIDAS in the European Union, NIST-DSS in the United States or ZertES in Switzerland.

Different kinds of electronic signatures

Diagram (for the case "qualified elecronic signature")

Advanced electronic signature 

For an electronic signature to be considered as advanced, it must meet the following requirements:
 The signer can be uniquely identified and linked to the signature
 The signer must have sole control of the signature creation data (typically a private key) that was used to create the electronic signature
 The signature must be capable of identifying if its accompanying data has been changed after the message was signed
 In the event that the accompanying data has been changed, the signature must be invalidated

Qualified electronic signature 
A qualified electronic signature is an electronic signature that is compliant to EU Regulation No 910/2014 (eIDAS Regulation) for electronic transactions within the internal European market. It enables to verify the authorship of a declaration in electronic data exchange over long periods of time. Qualified electronic signatures can be considered as digital equivalent to handwritten signatures.

Qualified electronic signatures use digital certificates, which are issued by accredited certification authorities. The certtificate and key are stored securely, usually on a smart card. To access the data on the smart card, the user must autenticate himself or herself, usually with a password or piece of biometric data. The certification authority also checked that the user is who he pretends to be, usually cross-checking with an official, state-issued document.

In addition to the points listed under "addvanced electronic signature", a qualified electronic signature also legally identifies the signer. before the authorities.

References 

Law
Computing
Internet