First you have to do is to make sure that the script file (.sh) is executable.
Make the script executable
In Nautilus, right-click it, select properties, permissions. Make sure that "executable" is ticked.
Now if you double-click the file you should be able to run the program.
Step 2: Create launcher
There two ways to do that.
The easy way
Install alacarte from Ubuntu software center or type
sudo apt-get install alacarte in a terminal.
Then open the "Main Menu" application and create a new launcher for your script.
A more complex way
You may create a .desktop file for your script by copying and modifying one of the files in /usr/share/applications.
A nice guide for creating .desktop files can be found here