Explain Host- and Domain Name in DD-WRT

Some ISP's probably expect you to set set up everything on a computer that is directly connected to the network through their modem etc. And perhaps because of security, or laziness some of them would then require your computer's name to match a specific one that they assigned to you. So this way if you connect to them through a router instead, this would allow you to set your router with the settings that they are expecting to see coming from you.