The following is a list of recommended mods that are available from the Steam Workshop only. Skyrim Nexus has many more mods available than are on the Steam Workshop, but the Workshop is probably more convenient for those of you who use Steam.
I currently have the mods sorted according to very rough, very general categories. If you have any suggestions for improvements or mods you'd like added to this list, please post a reply. If you'd like to maintain a list of mods available on Nexus, make the post and I'll set it as a sticky. You'll then be able to maintain it as you see fit.
Arthmoor's Bug Fix Mods
Arthmoor, apparently of some fame in the Oblivion modding community for his bugfix patches, has created three for Skyrim and each of its DLC packs. His patches fix numerous bugs, issues, and quests improving Skyrim's playability while taking particular care not to remove any objects or otherwise change the spirit of the game. Having run into a few issues myself, I decided to install his mods before anything else and they certainly do improve the quality of the game. Interestingly, Arthmoor continually updates all three mods and removes changes as Bethesda implements the fixes upstream.
Arthmoor's mods should be moved to the top of the load list, with the Unofficially Skyrim Patch first, the Unofficial Dawnguard Patch second, and the Unofficial Hearthfire Patch last. You should only install mods for the DLC you own. Click the "changelog" next to each patch link to read over fixes introduced into the game by the mod.
Unofficial Skyrim Patch / Changelog
Unofficial Dawnguard Patch / Changelog
Unofficial Hearthfire Patch / Changelog
Graphics/Texture Mods
The Workshop has tons of texture mods available and some of them are fantastic. The following list is a selection of the ones I use and recommend. Bear in mind that some of these mods have many alternatives, but the ones I've listed are the ones that work best for me.
Better Embers by Crashnburn. This mod doesn't change the look of embers dramatically, but it does darken soot and unburned coals inside the game's various firepits and braisers. If that's something that bothers you, Crashnburn's Better Embers might help. This mod may overrid the official high quality texture pack.
Enhanced Blood Textures by BakaKemono. If you want lots of blood, add a blood mod! This one works relatively well without transforming the game's blood effects into a cartoonish caricature of violence. This mod might override the official high quality texture pack--be warned!
HQ Snow Texture by Haugsnes. Skyrim's default snow texture may not be gritty enough for some and otherwise feel too "flat." The changes provided by this texture pack are subtle but improve the feel of the game. This texture pack may override the official high quality texture pack.
Improved Interior Lighting by Knightshade. Adds volumetric effects to interior lighting. May impact performance in some locales, particularly the Dragonborn's bedroom in Breezehome. Works well standalone, but installing a "Realistic Lighting" mod (there are several) will improve the ambience.
Lush Greens by Flip'd. There are at least two or three other "green" mods, but this one seems to have the best balance of performance and visual improvements.
More Grass by MadMonkey119. If you feel that the Skyrim terrain isn't busy enough and should have denser vegetation, install this mod. MadMonkey119's More Grass seems to work better than the others I tried and isn't overdone.
Pure Water by Laast. The Skyrim water is fantastic, but the streams and rivers seem to be missing something. Laast's Pure Water mod fixes this. Be aware that there are several other water mods and none of them look quite as good as Pure Water. Other mods tend to make Skyrim's water completely clear; Pure Water clears up the surface but adds a degree of of murkiness in standing basins (like small ponds and lakes) that looks more realistic.
Realistic Lighting and Colors by Flip'd. Fixes the ambient colors of interior lighting--and more! Use this with Knightshade's Improved Interior Lighting.
Mechanics/Gameplay Mods
Civil War Cleanup by joeThinkStudios. If you've completed the Civil War quest chain and have been somewhat annoyed that the locals of Skyrim have no capacity for rebuilding, this mod fixes that. It is recommended that you install this mod, run Skyrim, verify that war torn towns have been rebuilt, and then disable the mod (it's just a one time use mod). Pretty clever idea!
Help Out!
If you know of any mods you'd like to add to this list, let me know! Be aware that in order to be added, mods must be available for installation through the Steam Workshop.