Some quick questions here:

In what directory does the file that corresponds to the cat command exist? What would I type in the command line to find this out?

Also, what is the difference between the absolute and relative directory path?

Thanks in advance.

The cat command should be in your root bin directory (/bin/cat) - you can use the 'whereis' command to find the binaries and man pages for a command.

An absolute path leads with a '/' and designates a directory starting from the root structure (top of the filesystem). It will be the same no matter what. A relative path, however, varies, depending on when and where it is used. It generally looks something like 'bob/stuff' - no backslash to begin it. If you're in the directory 'home' and execute the command 'cd bob/stuff', it will look for the directory 'bob' and then 'stuff' starting in the directory 'home' [the one you're in]. If you execute the command 'cd /bob/stuff', it will look for the directory 'bob' and then 'stuff' starting from the root directory (/).

Thank you very much.

Try out the whereis command to locate command locations.