How to configure a client browser cert to be authenticated by a server? I currently am being asked to partake in a strange authentication format while working with a client, and was hoping for clarification.

The client is using HTTPS with client certificate authentication, therefore I will install their provided certificate within my browser as a form of authentication. 

As a second step, they are also asking that I create my own certificate and have a CA sign it, and then provide them with the certificate and CA so that the server is configured to authenticate with the client.

While I can create a certificate with openssl from the command line, have it signed, and send it to them, how is this authenticating/verifying my browser at this point?
You have to use the certificate (which will have your public key) which you had generated, to configure your browser. This certificate can be used by your browser to authenticate itself with server.