You can use the computer name in an active directory environment. Computer name will be resolve to an IP address on your internal DNS server if you have the host A record for that computer. And the client is pointing to that internal DNS.
In a workgroup environment netBIOS will do the job.
You can also use the HOST file, this file is the name to IP resolution that the computer first looks in before contacting the DNS server.
So yea, on the address box you can put in a computer name IP address or the URL. But I recommend IP address since it doesn't really need to be resolve.
If you put the URL, the DNS server must be able to resolve it to an IP address. Still the same process, but usually an IP address should be use so there isn't any problems.