Change ownership of external drive

If you have root access then fire up your terminal and write

su
Password: (enter your root password here.)

Now mount your HDD (if not already mounted). And type

chown -R "username_youwant" path_to_folder/folder_name

And now you're done!

It is generally a good practice to mount your HDDs in the /media or /mnt directory.

You go do this all through terminal, but I personally find it easiest to run my file manager (I use nemo but nautilus is default) as root and to change permissions graphically.

So open up a terminal and run gksudo nautilus.

Open the properties like you did in the screenshot. Since you're root now, you will be able to change it to your user.

This, however, is probably a better place to find a well explained answer.