Firewalls involve proxies. A proxy acts as a middle-man in a network transaction. Rather than allowing a client to speak directly to a server, the proxy server receives the request from the client, and then resubmits the request, on behalf of the client, to the target server. Each protocol or type of network transaction typically requires its own proxy program, and an administrator enables or installs specific proxies to determine what kinds of services will be allowed between the two networks.
Firewalls are not routers or address translators. Never does a firewall copy or forward a packet from the internal network to the external network, or vice versa. The internal network uses private address space. Neither side of the firewall knows about the address space on the other side of the firewall, and does not know how to route data to the other side of the firewall.
Some devices, such as the Cisco PIX, combine address translation with packet filtering. Like a firewall, this prevents the outside network from having knowledge of the address space on the protected network. However, aside from translating the addresses of the internal network, packets are forwarded as received through the unit, and no proxies are involved. This certainly improves security, but, strictly speaking, this is not a firewall.
It is worth noting that any good firewall will also employ packet filtering. This is done to protect the firewall itself from intrusion and to isolate intruders from the internal network should an attacker gain control of the firewall.
However, I believe it is important for network administrators to understand the difference between firewalls and packet filtering, so that they can properly evaluate different security options.