SharePoint and ADFS with Claims based authentication

Can someone break down how this works if I'm using a third party identity provider? Let's say I get to the login page and I select the third party identity provider as my authentication mechanism instead of Windows. Does this then use ADFS and check AD DS to see if the credential is valid? I ask this because the AD DS server has the AD accounts mapped to an x509 certificate, and supposedly the third party identity provider will not work without a smart card. But how is that? I can only presume the ADFS server is doing a lookup against the mapped account in AD DS?