Routers "route" traffic from one network to another. They can be used to connect different IP ranges/segments of larger networks together. Gateways usually refer to a router that performs the job of connecting the network to the internet. It's still a router, because it's connecting one network (the private network) to another network (the internet). All gateways are routers, but not all routers are gateway.

Modern gateway routers lets you share your cable or DSL modem's connection to the internet with all your connected computers. The gateway routers provides top-level security to your network with a NAT firewall that locks out would-be hackers, and you'll also be able to restrict access on any or all of the computers on the network. Using VPN (Virtual Private Networking), the gateway routers can be configured to allow remote access to the network by authorized users. Some of the popular gateway router brands are Belkin, Netgear, Linksys, D-Link etc.