A client socket does not listen for incoming connections, it initiates an outgoing connection to the server. The server socket listens for incoming connections.
A server creates a socket, binds the socket to an IP address and port number (for TCP and UDP), and then listens for incoming connections. When a client connects to the server, a new socket is created for communication with the client (TCP only). A polling mechanism is used to determine if any activity has occurred on any of the open sockets.
A client creates a socket and connects to a remote IP address and port number (for TCP and UDP). A polling mechanism can be used (select(), poll(), epoll(), etc) to monitor the socket for information from the server without blocking the thread.
In the case that the client is behind a router which provides NAT (network address translation), the router re-writes the address of the client to match the router’s public IP address. When the server responds, the router changes its public IP address back into the client’s IP address. The router keeps a table of the active connections that it is translating so that it can map the server’s responses to the correct client.