A normal server is an application or device that performs services for connected clients as part of a client-server architecture. A Normal server application can be defined as "an application program that accepts connections in order to service requests by sending back responses." while A "proxy server" is a computer that stands between a computer network and the Internet, sending and receiving information on behalf of the network user while preventing outsiders on the Internet from entering the network.
Not to be overly picky here Ashley, but you're only partially correct with your response. First, a Proxy server is well, a PROXY Definition of Proxy
It's a server that CAN but not always act on behalf of someone else/another PC/another server. It generally sits between the host and destination points in a connection, but that doesn't mean it's inline with the network configuration
There are several different types of Proxy servers and they do NOT always make the connection on behalf of you (see transparent proxy). Also. some just LOG/FILTER information for tracking purposes etc... and they do not ALL live between the internet and a network (reference anonymizer or open proxy). Additionally, proxy's do not necessarily prevent someone from accessing your network. I think you're attempting group a firewall with a proxy which is definitely NOT something you should do. They are definiitely two completely different technologies.
As for a normal server, this is a VERY broad term. A server in general can serve many purposes. It would be far better to say that a proxy server is merely ONE type of server.