GoldenCavalier: Here's where my lack of knowledge comes into play - I assumed all achievements for games in Galaxy were simply the same in game achievements, is this not the case? Can you give me an example?
MarkoH01: For the regular achievements to work you need a client such as GOG Galaxy or the Steam client. The client itself is handling the achievement list. So if you run such games without the client (which is possible with (almost) every GOG game, the achievements won't be shown or unlocked in any list at all which means that people who dislike using a client won't be able to use achievements at all. However there are some games (those on the list) in which the developer implements a special achievement section (you can see this in the screenshots posted for most of the contributions in this thread) and often also prgrammed popups that are created by the game itself not the client. Those games can be played with achievements even without using a client. The downside of these kinds of achievements is: you can't disable them at all and since there are some people who dislike achievements popping up during a game for those these ingame achievements are not positive at all.
Even a game that has ingame achievements might as well have normal achievements created by the client - sometimes you'll have both, sometimes you'll only have client based achievements and sometimes you'll only have ingame achievements. Sometimes the ingame achievements are identical to those created and listed in the client and sometimes (in rare cases) they are a bit different.
Thanks MarkoH01, I really appreciate you taking the time to explain. I originally came to GOG to replay old classics from my youth like the Ultima, Bard's Tale, and Might and Magic games without having to download things from abandonware sites and try and figure out DOSBox to use them. Not only did I find these games for pocket change here on GOG, I've found myself looking for more modern games a well, and achievements are something I've seen that definitely interest me. Thanks again for the explanation!