I am considering buying notebook that has wireless capability. The problem is my old 7 year old computer. The router I was considering has a larger telephone connection than my DSL ordinary telephone size connection. Would an adaptor solve the problem and what is required after plugging in?

The "larger telephone connection" you are referring to is called Ethernet.
This is an industry standard method of networking.
Basically with this router you plug your DSL in the smaller telephone socket and the Ethernet cable in one of the larger ones.
The ethernet cab;e then goes into a network card in your computer and from there you can access the internet.

Here's a whole bunch of info concerning ethernet networking.

Your wireless laptop will connect to the router via the wireless protocol. You may need to get an ethernet card or wireless adapter for the desktop.