To run Gish you will either need an archive of the demo (linked above) or an archive of the full game (available after purchasing, you will also need your activation key). To run Gish you just need to extract the archive somewhere and run the gish executable inside. Gish depends on a few libraries which may not be installed by default. These include the Open Audio Library (libopenal0a) along with its tool kit (libalut0), as well as the SDL libraries (which should be provided by the package libsdl1.2debian. Installing a packaged SDL game like Chromium should install any dependencies which are missing). If you are running a 64bit system make sure to install 32bit versions of these libraries (you can do this by simply extracting the package contents and copying the libraries inside to the /lib32 directory).
Double-clicking on the gish file should let you run Gish. However, if it doesn't then you can try running it from a terminal by opening a terminal, changing directory to the folder Gish is in and running ./gish. Any error information can be used to diagnose the problem (for instance messages about Ogg being missing would mean the libogg0 package needs to be installed).
There is a known issue with ALUT, the OpenAL toolkit, which can prevent the game loading with sound. This can be overcome by running Gish with a command similar to:
In the above case Gish is being run on a 64bit system, with 32bit libraries being stored in /lib32. Substitute the path to libalut.so.0 on your system for it to work.
On some installations, the game launches but reports the following error:
bt_audio_service_open: connect() failed: Connection refused (111)
In this case, the problem might be fixed by editing /etc/openal/alsoft.conf (but don't forget to make a backup of the original!) and adding the following configuration directive:
WARNING: This fix is not an official recommendation, it's just a way to get things to work in some installations. Be warned that editing system configuration files in this fashion might cause other issues, so if something else stops working, revert your changes.