What would be the difference between plugging a switch into your router, and plugging another router into the main router?
ok so you are mixxing up a TRUE router and a SOHO router a soho router combines a router a switch and a wap a router can ONLY connect two different networks a soho router functions as a switch as well so it can handle connecting to the same network with no issues, it just goes into "switch" mode a pure switch will work better but a high end SOHO will be fine.
Switches only provide more opportunities to hard wire more computers to it to the LAN. Nothing more! In respect the "routers" you need to understand that what you are calling Router A, is actually a WiFi Modem in most cases these days. They not only broadcast a wifi signal that is typically rated for a 300 foot radius but also have connectors to accommodate a actual router or additional computers via hard wire connections. Typically a modem will have a connector for the primary computer in the LAN, and an additional 4 connectors for other computers. However, a daisy chain of routers and/or switches is not the way to go. A better option would be to use a Frequency Range Extender as opposed to using an additional router to get the signal to areas with little signal. What the Range Extender does is pick up the weak signal and then boosts and rebroadcasts the wifi signal at a much higher strength so that your ability to connect to the internet and realize good page loads, etc. is normally really good. I purchased a Range Extender on sale from $54 to $18 on amazon.com about 2 years ago and it works great. Hope this helps.
When you connected the router from LAN to the WAN side of the other, you created 2 networks. Things one on segment cannot talk to the other easily, like you have to tweak lots of firewall rules and network discovery is broken, like printer discovery. If you went LAN to LAN AND turned off things on the second router, then you both extend the wired side AND the wireless side, and everything is on one network. This is AP mode, but a waste of the routing engine and firewall software of the second router. When in AP mode, the AP is just a dumb switch for both wired and wireless. A switch is just a wired network extender, and used to get more ports on a network.
Well, there's a difference between a router's WAN (Wide Area Network) port and its LAN (Local Area Network) ports. So, let's start there. A SOHO (small office, home office) router uses a WAN port to connect the outside or public world (or Internet) to its inside world (or local network). So, to a SOHO router, anything outside of the WAN port is foreign and anything inside is part of the local network. Knowing this, we can say that connecting 2 SOHO routers, where router-2's WAN port is connected to one of router-1's LAN ports will create 2 separate and unrelated networks. This is EXACTLY what you'd do if, say, your home-office network had to remain separate from your home network (where your wife and kids connect). So, no, a connection like this does NOT create a single LAN network. It, in fact, creates 2 LAN networks. Now, let's look at your "switch" example. In your scenario, we connect 3 devices to 3 of the router's LAN ports. Then we connect a switch to the 4th LAN port and connect more devices to the switch. This is EXACTLY what switches were created to do! Take 1 port and let you expand it to many. And since everyone is plugged into the router's LAN ports they are all able to "see" one another as they exist in 1 large network.