I know this gets complicated.

My husband's laptop does not have wifi--- so he plugs directly into the router with a cable, like Jo describes-- a long long cable from the next room. Our laptop upstairs does have a wifi and gets our DSL from the router.

We have a Linksys router 24GHz (which has 4 plugs for cables in addition to the antenna actions) which was about $45 at Best Buy.
Previously we had a NetGear one which was very problematic and caused us much grief with its on and off temperament and erratic functions. Tiger Direct often has very good prices too, online.

If you have DSL your modem will only have so many plugs.
For example, this computer is directly plugged into our modem.
When the old router failed, I could still get onto the internet with this desktop by unplugging the router and going directly into the modem. You for example would plug the modem into the router, then using a cable you plug your desktop into the router, and then the router will send the signal to anywhere you are within its range, thru the air like a radio. Because our router is located in a room facing the yard, we get its signal outside at our picnic table. It is actually weaker upstairs.
What I still find confusing is when the modem fails. Our AT&T DSL sometimes goes down. When that happens you have to reboot your modem, router and computer in a certain order, to reset it all. I still get confused doing that, so hubby does it.

Newer laptops now have 3G cards that will for an extra fee, put you into a cell phone type linkup. That means you go thru a cell phone tower, and don't need a modem like with DSL. There is a monthly charge for this like a cell phone.

If you are getting a newer laptop I'd suggest you get a GOOD battery. At least a 6cell. Our Dell battery was poor and now dead, and we wish in retrospect we had considered a better one at the time. Now we have to replace the battery and it will be big bucks!