When normal (non-root) users log in to a computer locally, they are given two types of special permission; they can run certain programs that they would not otherwise be able to run, and they can access certain files (normally special device files used to access diskettes, CD-ROMS, etc.) that they would not otherwise be able to access.
Since there are multiple consoles on a single computer, and multiple users can be logged into the computer locally at the same time, one of the users has to ``win'' the fight to access the files. The first user to log in at the console owns those files. Once the first user logs out, the next user who logs in will own the files.
In contrast, every user who logs in on the console will be allowed to run programs normally restricted to the root user. By default, those programs will ask for the user's password. This will be done graphically if X is running which makes it possible to make these actions menu