A network server is just a generic term, it refers to *any* machine that provides *any* services (printers, files, database, etc.) to other machines over a network. One of those services might be a proxy.
A proxy is a service that acts as a go-between your machine and the intended destination machine. That proxy could exists for many reason, such as provide authorization to the destination resource, filter the content that's returned, or simply hide your source IP address from the destination machine (iow, the destination machine only sees the proxy's IP address, not yours).
The other common use for a proxy server on a network is to speed up websurfing AND reduce internet traffic. The proxy will act as a go-to for the web browser when it requests a webpage. If the proxy has the page stored on it because it has been retrieved before, the proxy sends that info, and the internet isn't used - FAST! If the proxy doesn't have the page, then it requests the page from the internet, stores it, and sends it on to the computer that has the webbrowser.