Will a router installed in my laptop connect me to the internet sans any external connections?

A "router" is an external device, not one that is installed into a computer.

In order to connect to the internet, you need to subscribe to a service. That can be broadband (Cable, DSL, satellite) or narrowband (dialup via a phoneline).

A router simply allows for the connection to your Internet Service Provider to be split to multiple destinations (for example, if you have multiple computers in your house). Routers are also used to create a Wireless Network.

The first step is to subscribe to a service. If the service is broadband (cable or dsl) the provider will provide you with a modem. Your computer can then be connected directly to the modem, or a router can be placed in between the computer and the modem if you wish to split the connection to multiple destinations.

A router will only gets internet connection from active internet from a modem like a cable or a dsl connection, the router will just share it to the computer connected to it...