Some routers have modems built in. When an ISP assigns you with a public-facing IP (what all the websites see you as), it's generally referred to as the WAN address or WAN IP.
Many routers have NAT (don't worry about it for now), and indeed unless you have multiple IPs assigned by your ISP, you will need NAT to have private network addresses assigned to the computers in you network (don't try to take all this on as a heavy learning issue).
When you configure IPs on the computers on your network, you get asked to assign the 'default gateway' for each one. This is the IP that connects your computer to your network. In most cases it is your router. This is generally referred to as the router's LAN address or LAN IP.
So the router can have a LAN IP (internal) and a WAN IP (external). One is used to connect to the other devices on your network, while the other is generally used to communicate with the world.
You could have the following setup though:
Computer (IP address) - Router 1 LAN to LAN - Router 2 LAN to WAN - Outside World. It would be perfectly feasible to have multiple computers directly connected to either router in this setup. Sometimes it's handy for isolating or separating groups of users / devices. One idea might be to have a server connected to router 2 for the world to access, while router 1 secures the PCs on the network from external access.
LAN to LAN generally refers to internal connections, while LAN to WAN refers to the gateway to the outside world. Neither are better or worse than the other. It's more a case of applying devices to suit your needs. You really want the most secure, but least complicated setup.