Basically a digital ID or certificate is an electronic signature, which is made secure using encryption techniques. Basically a math function which has an inverse is used to create a certificate. You have the inverse function secret and you make public the regular function. People use this regular function to send to encrypt e-mail and send it to you. Since you are the only one with the inverse function, only you can decrypt the message.
Signatures work exactly the same way.
You create a signature using your inverse function, and then the person who receives this digital signature uses your regular function to decrypt the signature and verify that it is you.