What can an attacker do with a stolen website certificate and key?
What should the website admin do upon learning this?
Attackers in possession of the private key can:
They should contact their Certification Authority to revoke the CA, and get a new one (with a new key part). Once revoked, clients checking for revocation (via CRL or OCSP) shouldn't accept the old certificate. (Clients that do not check this will still be vulnerable, until the expiry of the certificate.)
Telling users that this incident happened isn't a bad idea, at least to encourage them to check their browsers check for revocation (and perhaps take necessary action, should they suspect some of their communication have been vulnerable).
Note that, if the attackers have recorded past communications using that certificate and that a non-PFS cipher suite was used at the time, the attackers should be able to decipher those past communications with the private key.
If an attacker uncovers your SSL private key, you are minimally vulnerable to the following attacks:
1) Traffic Eavesdropping
To the attacker, everything being sent over SSL can be decrypted and should consequently be thought of as cleartext. This means passwords, credit card numbers, and other Personal and Private Information is vulnerable to being either harvested or leveraged against you as the attack escalates.
Part of the reason to use an SSL certificate is to uniquely authenticate yourself to the clients connecting to your server. If the private key is stolen, a hacker can create a Man-In-the-Middle attack where data flowing either from the server-to-client or client-to-server is modified in-transit. This could be done to ask a user to reauthenticate (and thereby surrender their password), ask for a credit card number, or implant malware into file downloads.
As you can see, if someone has compromised your SSL certificate, they can quickly escalate the attack to either gain unauthorized access to your system or attack you or your users.
How to Recover
If a private key compromise is suspected, one should add the SSL Certificate to the Certificate Revocation List (CRL). This will alert other participants in the Public Key Infrastructure (PKI) that the certificate in question can no longer be trusted. In order to do this, you will usually need to login to the account you created with the Certificate Authority (CA) who issued the SSL Certificate or otherwise notify them of the suspected breach. Godaddy provides instructions for this online, as does Network Solutions.
With the SSL certificate revoked, you will need to again go through the process of generating a new certificate and private key in order to re-enable SSL services for your web site or application.
If your site has been breached, I would recommend recovering in the following steps: