Well depending on what you mean by this, yes. You can. Have one cable, attatch one router to one end and the other to the other end and now they are on the same line. But, if you're talking about splitting a cable or something, I'm not sure about that.
Yes, you can have two routers connected to the same internet connection. However, for a home type of connection, this is usually a problem since internet providers only allow one PPPoE connection (DSL), and cable proviers may limit the number of MAC addresses conecting to the line.
However, in a corporate network, the internet connection may be plugged into a switch, which then has two routers connected to that switch. From each router, the network segments continue to be connected.
Please explain what you are trying to accomplish so that a proper recommendation can be given to you.
@JorgeM Basically I need this Because I'm trying to host my own home webserver and I need to have another router instead my old one to have full functionality. I do not want to throw the old one away but I need to have a new router with only one connection which is to the server
So, you could have a perimeter router that connects to the internet, then just plug in additional routers into that perimeter router. However, if you are not familiar with routing protocols, this will be challanging for you.
You mentioned a web server. I would suggest that you consider hosting the web server with a provider instead. If you are trying to set up a web server to sell websites, I encourage even more. Keep in mind that at home, you typically do not have a fault tolerant, redundant set of systems. Hosting providers have already invested in this type of infrastructure.
There are a lot of providers out there that will sell you a service, which allows you to "re-sell" the service making you a web hosting provider for your customers.
Do some research before you move forward with setting this all up at home.
I have tried doing the same thing. The only way i can think of, to provide and internet connection the the 2nd router, and if you cannot use more than 1 DSL connection, use a cross-over etthernet cable (not a standard ethernet cable) and then connect your web server to the second router, therefor giving you the functionality of the first router, on the second router, but only using 1 DSL line.
Yes, what you get looks like a 'series' circuit where internet is delivered by modem, modem goes to router 1, router 1 goes to router 2.
In this case, I don't see the need for multiple routers. All you need is a router that suports a dmz connection of does port forwarding in order to support at home web servers. DD-wrt does this well, but most consumers routers do this anyway.
I would not use a home network for any serious web work. You can easily get a hosted service for $10 a month. IF you know what you are doing, even VPS service can be had for $5 a month.