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What is a digital signature?
A digital signature is an extra chunk of data send along with an (encrypted) message which tells the receiver that the message is coming from the one who claims to have sent it. It ensures that the message has not been tampered with by a man-in-the-middle, a person who spies on the communication line and intercepts messages. You also have a proof that the sender is the one who holds the private key of a public-private key pair.

It goes as follows:

 
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But who says this public/private pair belongs to me? I could have identified myself as someone else, generate such a pair and distribute the public key to a sender who believes I am someone else and trick him with this above process claiming to be that someone else...

That's where digital certificates come in. Check out What is a digital certificate?


Further Information
Author of answer: Joris Van den Bogaert

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