If your friend is using a Modem then yes it is normal to have to dial up each time you switch on. Most of these modems are powered by the USB port and you'll see that when the PC goes off the modem also goes off.
Your WiFi (Wireless Router) maintains a permanant link to your ISP and is powered from the mains so doesn't go off when the PC does.
Modems are usually only for the PC that is connected so they don't need to keep the connection alive when the PC is off.
Routers may also be need by Wireless enabled laptops and PCs so need to stay on when the PC isn't on.