Operating as a router allows a linux machine to accept packets on one interface and transmit them on another. This is the nature of a router. The process of accepting and transmitting IP packets is known as forwarding. IP forwarding is a requirement for many of the networking techniques identified here. Stateless NAT and firewalling, transparent proxying and masquerading all require the support of IP forwarding in order to function correctly.
The sysctl net/ipv4/ip_forward toggles the IP forwarding functionality on a linux box. Note that setting this sysctl alters other routing-related sysctl entries, so it is wise to set this first, and then alter other entries. Frequently, an administrator will forget this simple and crucial detail when configuring a new machine to operate as a router only to be frustrated at the simple error.
The sysctl net/ipv4/conf/$DEV/forward defaults to the value of net/ipv4/ip_forward, but can be independently modified. In order to allow forwarding of packets between two interfaces while prohibiting such behaviour on a third interface, this sysctl can be employed.