How to run .sh file
copy the file to the desktop (for convenience)
Right click on the file and select properties
Under permission tab check allow executing file as program
double click on the file and chose run in terminal if it is not a gui program)
By default the file manager may not show that option and open the file in gedit instead. In that case change the preference of the file manager at: Edit-> Preferences -> Behaviour to "ask each time" or right away to "run executables"
Command line
cd /path/to/file
chmod +x filename.sh
./filename.sh
Comment below if it wont work :)
If it still won't work, despite having allow executing file as a program ticked, when you double click on the .sh file, and it launches gedit, navigate to the folder with the script.
Once you are in the correct current folder for the script, you can run the script like this:
sudo ./filename.sh
If that doesn't work you may try
sudo bash filename.sh
From FAT partition
Follow this How do I execute a file from a FAT USB drive?
Command line
Apart from what liquid had recommended, usual procedure is to change permissions to make this file executable:
chmod +x file.sh
And then execute this file:
./file.sh
GUI
The same can be achieved in Nautilus file browser using context menu by changing Execute flags in Properties -> Permissions.
You can also use . tricks, with the suggestion of other answers.
For example:
chmod +x filename.sh, assuming you have already gone to the path to file
then run the file with any one of these command:
sh filename.sh
./filename.sh
. filename.sh