If you want to take your notebook out and about, you would either need a USB dongle to connect to the mobile network, or a wireless hotspot to connect to. There are lots of them about, but the dongle is probably your best bet if you're out in the wilds of nowhere (but still with an available signal).
If you're going abroad, then a dongle bought here probably wouldn't work (I've never tried it), and you may have to rely on local wireless hotspots.
A router would be only used in your home to connect to your internet provider - it can be either wired or wireless.