Contrary to what their most common use would lead you to think, su and sudo are not just meant for logging in (or performing actions) as root.
su allows you to switch your effective user ID with that of someone else. For this reason, when you type su, the system needs to verify that you have the credentials for the target user you're trying to change into.
sudo is a bit different. Using sudo allows you to run certain (or all, depending on the configuration) commands as someone else. Your own identity is used to determine what types of commands sudo will run for you under someone else's identity: if you're a trusted user (in the sense that the sysadmin trusts you), you'll be allowed more free rein than, say, an intern. This is why sudo needs to verify your own identity rather than that of the target user.
As for what you were trying to do, just sudo su root, or even more simply sudo su and type your regular user password. This assumes the sudo configuration allows you to run su with escalated privileges.