I need help finding a laptop for my husband.

Does the router question relate in any way to the laptop question because we have a forum for Networking.

I suggest you post there with something with little less general intent as simply asking what a router does or how it works isn't probably what you need to know in practical terms.

I think the lenovo thinkpad units are generally well designed. Anything current should be strong enough and will fit your budget:
http://shop.lenovo.com/us/landing_pages/promos/search/e...|semd|se|google|ThinkPad_Laptop_Special_Offers|Lenovo_SiteLinks|NE_Lenovo_Only|100000&&ne_ppc_id=1019&ne_key_id=9728652&ne_sadid=6290414024

A router's function is to take an input and distribute it to where it needs to go.
A typical use would be to connect an input to a cable modem, and then distribute the signal to several connected pc's.
A router often has a wireless in/out capability also.

+1 for simple and basic explanation

A simple analogy for a router would be a post office, The information (mail) is delived by truck (cable modem) to the post office (router) and the post office then decides who the information is heading to and sends it off, same thing happens in reverse when sending stuff like emails. Many routers have wireless capability which does the exact same thing just without the hassle of wires.

Congrats on your husband getting a new job, driving 8 hours a day and staying in a hotel will be tough. As long as the laptop isnt dropped/ have stuff spilt on it, any current laptop will suffice.