By default, the root account password is locked in Ubuntu. This means that you cannot login as root directly or use the su command to become the root user. However, since the root account physically exists it is still possible to run programs with root-level privileges. This is where sudocomes in - it allows authorized users (normally "Administrative" users; for further information please refer to AddUsersHowto) to run certain programs as root without having to know the root password.
This means that in the terminal you should use sudo for commands that require root privileges; simply prepend sudo to all the commands you would normally run as root. For more extensive usage examples, please see below. Similarly, when you run GUI programs that require root privileges (e.g. the network configuration applet), use graphical sudo and you will also be prompted for a password (more below). Just remember, when sudo asks for a password, it needs YOUR USER password, and not the root account password.
Ubuntu is also setup initially with the first user having the right to become root through SUDO by being a member of the Admin group. Subsequent users that need root privileges simply need to be added to this group.
It is advisable to NEVER change the root password from it's default hash for many reasons. If you need persistent root access simply use sudo -i.