I am trying to understand the relationship between the client and server in the context of an SSL connection. Am I correct in understanding that the fact that the same certificate authority (me - in example below) sign both server and client certificate makes that they can communicate. Thus, that the server only accepts communication when client authenticates with client certificate signed by the same CA as the server certificate, and this is essential to the idea of an SSL connection?

(script underneath comes directly from http://blog.nategood.com/client-side-certificate-authentication-in-ngi)

# Create the CA Key and Certificate for signing Client Certs
openssl genrsa -des3 -out ca.key 4096
openssl req -new -x509 -days 365 -key ca.key -out ca.crt

# Create the Server Key, CSR, and Certificate
openssl genrsa -des3 -out server.key 1024
openssl req -new -key server.key -out server.csr

# We're self signing our own server cert here.  This is a no-no in production.
openssl x509 -req -days 365 -in server.csr -CA ca.crt -CAkey ca.key -set_serial 01 -out server.crt

# Create the Client Key and CSR
openssl genrsa -des3 -out client.key 1024
openssl req -new -key client.key -out client.csr

# Sign the client certificate with our CA cert.  Unlike signing our own server cert, this is what we want to do.
openssl x509 -req -days 365 -in client.csr -CA ca.crt -CAkey ca.key -set_serial 01 -out client.crt

server {
    listen        443;
    ssl on;
    server_name example.com;

    ssl_certificate      /etc/nginx/certs/server.crt;
    ssl_certificate_key  /etc/nginx/certs/server.key;
    ssl_client_certificate /etc/nginx/certs/ca.crt;
    ssl_verify_client on;

The short answer is No. These are two separate aspects. Here:

ssl_certificate      /etc/nginx/certs/server.crt;
ssl_certificate_key  /etc/nginx/certs/server.key;

You are configuring the server certificates which need to be trusted by the client.

And here:

ssl_client_certificate /etc/nginx/certs/ca.crt;

You configure the certification authority to verify your clients' certificates against.