Security :: Web Client Authentication Through PKI And CACs?

I'm working on a work project related to Web (Client) authentication and DOD Common Access Cards. But I'm having difficult getting the details about what happens on the CAC side of things.

I familiar with the PKI system as it applies to e-mail. (Correct me if I err, of course.) If you want to sign an e-mail (i.e., so it can be authenticated by the receiver) you use your private key to add a digital signature to the message. Then, the receiver uses your published public key to determine if the digital signature is valid, i.e., was created using your private key (even though the receiver never actually has access to your private key).

So... my questions:

1) When a person with a DOD CAC visits a CAC-enabled web site, and the server grants access after the CAC is inserted, is the authentication process fundamentally the same as what happened with the e-mail authentication?

2) If the private key is used in this process (it would have to be, correct?) is the signature created on the CA Card electronics (i.e., the private key remains on the CAC)? Or is the private key copied onto the computer, which uses it to create the signature?