It sounds like you understand this now, but just to make sure - “tarball” is an informal term for archived files in the tar format. These are very widely-used to distribute open-source software. Unlike .zip files, the .tar format doesn’t actually do any compression, so it’s common to compress the file after archiving it, which is how you end up with a .tar.gz or .tar.bz2 file. You very likely want the 64-bit version of the “Generic Linux Binaries for x86”. You can safely ignore the “GPG” link, which is useful for folks who want to verify that the file isn’t corrupted or tampered with by the time it makes it to your hard drive.
it would also help if you shared how you usually start Julia. One tricky thing here is that there are many ways to install and run Julia, and experienced folks tend to have their own ideas about how they like to do things. That can make it more confusing to less-experienced folks, as you may get advice that doesn’t pertain to you.
Running in-place
It sounds like your current julia installation is living in a folder in your home directory called “julia-1.0.0”. Do you usually run Julia from within that folder (e.g. by double-clicking it in your file navigator, or from the terminal)? In this case just copying the folder and doing what you usually do should be fine.
Using Juno
If you use Juno than you should be fine just unzipping the tarball and copying the julia-1.0.3 directory wherever you want, and then going into the Juno settings and setting the path to the Julia executable to the new folder. Then you can delete the old 1.0.0 folder if you like.
Modifying your PATH
As an alternate method, you may also hear people talk about adding things to your PATH variable. This is a unix variable that determines where things are searched when you try to run them from the terminal prompt. If you open a terminal in your home directory and type julia, the terminal looks in the directories listed in your PATH to see if it can find a julia executable. If you add your new julia-1.0.3/bin directory to your PATH than you’ll be able to run julia from any directory in your terminal. There are multiple ways to do this, so I’ll leave that as a dangling pointer to be possibly pursued in the future.
Symlink to your PATH
Yet another way that is popular, which is to create a symbolic link inside a directory that is already in your PATH (you can type echo $PATH in your terminal to see which directories those are). A symbolic link is a small file that lives in one place, but points to a file somewhere else. So you can put your julia-1.0.3 directory wherever you want, and then place a symlink in your PATH that points to the julia executable. /usr/local/bin is a pretty common directory to put these sorts of things. The command to do this would be something like:
sudo ln -s /home/gaussia/julia-1.0.3/bin/julia /usr/local/bin/julia
here sudo gives you superuser permissions (which are likely necessary to modify /usr/local), ln -s is the command to create a symbolic link, and the two paths are the source and destination paths.
Hopefully this is clarifying more than confusing.