Installing mods for Amnesia is usually a very easy process. Always look at the included README file, but generally to install mods for this game, all you have to do is drag the mod's folder into the Amnesia\custom_stories folder. Full conversion mods are a bit different, but still easy. They usually just require dragging many folders/files into your main Amnesia folder.
Disponentia by Mackiiboy. One of the most original and interesting custom stories, based on a true story and a real haunted mansion. Very innovative, I consider it to be the scariest Amnesia mod.
Emma's Story by Evil Armor. It even recreates the main hall and dining room of Resident Evil.
Amnesia Hill: Harold's Story by VengeanceFilms. It's a custom story inspired by Silent Hill, but more directly than all the others. Complete with Silent Hill references, similar atmosphere, and some of the same sound effects. It's a very well made mod and very long, there's essentially two long campaigns for it.
White Night by Tanshaydar. The most famous Amnesia full conversion mod, very well received and very original.
Through The Portal by DamnNoHTML. The most wild, mesmerizing Amnesia mod. Amazing otherworldly experience.
Mental Memorial by DnALANGE. A full conversion mod with distinct atmosphere, new monsters, new gameplay mechanics, new sounds, and is just very well put together.
La Caza by Team Ninja-Samurai. An award-winning, highly sought after mod.
Abduction by Anxt. The most true to the original game.
The Attic by Everlost. A three chapter series, ranks #2 as scariest Amnesia mod.
Out of Time by Custommapper. One of the most innovative mods, full of potential. It could use some improvements in level design/bug removal though.
Wake by DamnNoHtml. A short but very unique custom story, from the same maker as Through the Portal.
The Four Horsemen by Titan Core. An interesting tale about these biblical figures. The mod contains very good level design, at times reminding me of dungeons found in D&D games. Voice acting is above average, and overall it's well put together.
The Things in the Night by DoomGod Collab Productions. One of the longest and most impressive custom stories. Unpredictable and an overall amazing experience.
There are plenty of other enjoyable campaigns, but these are the ones that stand out the most. Moving on...