I keep getting authentication failure messages, root password?

Please advise

Impossible to advise without having any information.
What linux distribution are you running?
What are you trying to do. login etc.

I received an operating system disc with a Linux magazine I purchased. "Fedora 16"
I installed it on my hard drive a while back.
When I try to go to my other back up hard drive I get a message, " Authentication is required to mount the device" password for root? authenticate?

When on line using utube , message "install missing plug ins"
Message "additional pliugins are required to display all media"
when I push "install plug ins" , then the "no suitable plug in was found" error message is displayed.

when I try to download updates, error message, "authentication is required to consider a key used for signing packages as trusted."

I have no clue what the "root password" would be or how to obtain one.

You should of been asked to enter a root password when you installed Fedora.