Question: What additional privileges does the root user have over a standard user in Linux?

Answer: 
The root user can do pretty much anything that the hardware allows. The root user can write directly to hard drives without going through the filesystem. Root can modify the kernel via modules. The root user can also circumvent any security policy of the system software: he can change ownership of any files, set up privileged ports, write to privileged filesystem locations, gain extra scheduling priority levels, etc. The root user can also spy on any other user in the system.

Regular users can generally only manipulate their own processes and their own filesystem subtrees. They tend to be forbidden from manipulating the system's hardware, security policies, and from meddling with other users' files and processes.
root is the system administrator, and they can do whatever they want to the system, which can lead to data corruption if the account is not used wisely. Since it appears you are new to Linux, I would recommend you not use root until you get more experience with Linux. To access an administrator/root command prompt from a standard users command prompt, type `su` (I believe most distros support this) and then hit enter.

*Beware when using root, as you can really mess up your computer.*