According to the U.S. Federal ESIGN Act, Electronic Signatures are defined as: "Electronic sound, symbol, or process, attached to or logically associated with a contract or other record and executed or adopted by a person with the intent to sign the record."
Basically, an Electronic Signature is the equivalent of your hand written signature digitized and can be used to confirm content within a document, or the terms of a particular document.
A digital signature is a type of electronic signature that offers more security than a traditional electronic signature.
When you sign a document with a digital signature, the signature links a "fingerprint" of the document to your identity.
Then that information is permanently embedded into the document, and the document will show if someone comes in and tries to tamper with it after you have signed it.
Digital signatures offer tamper evidence, independent verification and a strict adherence to ISO standards.
Because that information is embedded in the document, you donâ€™t need to check back with the vendor if you want to verify that the signature is still secure.
Thatâ€™s a big benefit if you donâ€™t want to be tied down to one vendor over the life of your documents.
On top of that, more countries around the world accept digital signatures because they comply with international standards for security.
See also the Wikipedia definition for digital signatures