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).