Since, from your forum user registration date, which states you joined this month, I assume you are new to GOG and how it works, I'll try and give you a little rundown on the basics:
GOG is proudly DRM-free, and the games you buy through their service are *yours* to keep and do whatever you please with them. As of now, GOG doesn't have a client, like that of Steam, through which you can launch and play games. What you do is download the installer via your GOG account library and install the game in your computer (you can also save a backup of the installer, if you so desire). Think of it as installing and playing games before the "digital revolution": there's no GOG preset folder for GOG games, you're free to install them anywhere you like in your computer, and the game is usually a stand-alone product. You don't launch it via any client. So, as soon as Tyrian -- or any other game -- is installed in your machine, it should appear in your Start Menu (assuming you're using any Windows OS) under Start -> All Programs -> GOG.com -> Tyrian 2000. A shortcut of the game is also created in your Games folder (on Windows Vista, 7 and 8). If you wish to create a desktop shortcut for GOG games, go to the folder in which you install the specific game, copy the GOG launcher or the game's executable and paste it on your desktop. GOG games have desktop shortcuts by default when you install them, so if you don't have one for Tyrian, chances are you didn't install the game at all or you unchecked the "Create desktop shortcut" option when installing the game.
GOG games, especially oldies like Tyrian, are completely DRM-free and do *not* require any kind of internet connection, so, rest assured, there's no need whatsoever to log in to your GOG account in order to play them. They're all permanently in "offline mode".
I'm sorry if this sounds a bit confusing -- it really isn't, it's just that English isn't my native tongue, and I apologize in advance for any confusion my explanation might have caused you. Feel free to ask for any clarification, I'm sure myself or any other community member will be glad to be of help.
And welcome to GOG! :)