Welcome to Linux.
"Root" is an account on your system. When you log in, you can use "root" for the username and then enter the password you specified for root during setup. The
root user doesn't have any limitation on privilege and as such can modify any file, run any program, and programs/processes run as
root can do anything they want.
Thus it's dangerous to run as
root unless you need to, so the usual practice is to have a standard user account for day-to-day use and then elevate to
root when needed to perform an administrative action, such as the one you are trying to do. (If you've come from a Windows background,
root under Linux lets you be far more destructive to the system than Windows does under an admin account.)
Files also have permissions and owners - permissions are divided into "read", "write" and "execute." There are three sets of these permissions, one for the file owner, one for the group the file belongs to, and one for everyone else.
root or the file's owner, or an account that belongs to the "group-owner" can always change permissions of a file. (I won't get into the nuances of permissions here but
man chmod is a good starting point for more learning.)
Since the Debian distribution uses
/etc/network/interfaces to define network interface information for the
ifdown commands, it's considered important - and therefore protected by only having "read" permission for anyone other than the owner or "group-owner".
So, to get write access to this file, you need to either:
1) login as root on your system and then make the changes you need
2) use a method of running a single command as root - the
sudo command makes this rather simple - so a
sudo vi /etc/network/interfaces should allow you to edit it, once you enter your password. If you are running a graphical desktop, look for the "Terminal" or "xterm" application.
Check out the Debian wiki for a reference to the options in /etc/network/interfaces: http://wiki.debian.org/NetworkConfiguration