A router is a device that forwards data packets across multiple networks. Android devices are interacting most commonly with Wifi routers, but they can also have a router role, and other physical layers are possible (bluetooth, usb, ethernet).
In general, routers perform the data "traffic directing" functions on the Internet. Most Android devices interact with home routers, which bind together a home or small business network to the uplink network of an ISP (internet service provider).
Often the Android device acts as a router, most commonly if it shares its mobile network uplink with other devices. This happens most commonly over usb, bluetooth or wifi.