Wifi = wireless.
Basically a router need's to have a connection to the internet, either via a dialup modem, or a cable broadband modem or dsl/adsl or fios modem/device.
What a wireless router does, is spread or broadcast the internet network over a physical area, like a radio broadcast.
I do not know much about kindle.
But once your router is setup, so that other devices can use/access the internet, then you need to set that up for the kindle, presuming it has a wireless/wifi capability.