Let's take this a step at a time.  Switches are network neutral, and this includes the switch in your Linksys router.  Most basic switches do not have their own address. They just connect addresses on the same network.  For what we do, the network is defined by the first 3 groups in the IP address. So 192.168.1.x  is the network and the x represents a specific address sitting on that network.

Because of this a router is needed to link different networks.  It will allow something on the 192.168.1.x to something on the 10.0.0.x (another private network.)  The other thing a router can do is connect a private address to a public address.  So your address can find yahoo.com (actually 66.94.234.13.)  It does this by using a process called NAT.  What happens is that it removes your private address from the packet and replaces it with the public address on the public side of the router. It also maintains a table so when an answer comes back to it, it knows how to forward to the device that sent it.