Note: Pretty much any mod found within Steam Workshop can also be found on SkyrimNexus.com. The opposite however is not true, and there are a lot of great mods there (Nexus) that haven't made it to Steam for whatever reason.
Installing mods from Nexus can be done via their own mod manager app, manually adding it with the app (using the + button) by pointing it to the downloaded .zip, or by upzipping and dragging files old-school style into your Skyrim directory per the mods instructions.
I will update this post if I discover a Nexus mod is actually now on Steam or under a different name.
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*** UI ***(Steam/Nexus)"SkyUI"http://skyrim.nexusmods.com/mods/3863a complete overhaul of your inventory, merchant interaction buy/sell, shouts, spells, you name it. Lets you sort items by A-Z, by weight, by equipped, and so on. Phenomenal mod. Requires SKSE Loader for full features, though IMO that is optional. SKSE lets you use a search box within your inventory, so you can do quick string searches to filter out inventory items on the fly. To use SKSE, unzip it into a temp directory, then drag it all into your /steam/steamapps/common/skyrim directory. Once Steam is open, you will now use SKSE_Loader.exe to start your game, so modify your steam shortcut for it or create a new icon on your desktop. If you don't want to do that and don't give a shit about Search, SkyUI will work the same, save for a nagbox in the corner telling you SKSE isn't loaded.
(Nexus)"Categorized Favorites Menu"http://skyrim.nexusmods.com/mods/4862Favorite/Star items, potions, spells, shouts, weapons to your heart's content - go nuts! This mod auto-sorts and arranges your Favorites menu (Q key in game) by type, subtype, and so on. Fully compatible with SkyUI. Also uses SKSE for customized equipment set macros, and also rearranging the default column layout. Again, SKSE is not needed if you do not care about these features.
"Better Sorting"http://skyrim.nexusmods.com/mods/2730Fantastic item type and quantity grouping. All mined ore now appears as "Ore - Iron (5)", "Ore - Gold (3)", and so on, making for A-Z sorting and an overall cleaner view of your inventory and merchant interactions. Fully compatible with SkyUI and Categorized Favorites Menu. Also comes with completely optional choices to modify (lessen or make outright weightless) the weight values of common items such as ore, food, soulgems, etc. Default install keeps these at their regular weight.
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*** Quality of Life ***(Steam/Nexus)
"Cartographers Map Markers"http://skyrim.nexusmods.com/mods/15489Adds over 180 Map Markers (you must first discover them, as usual) including 73 In-City Inns, Stores, Player Homes, and Important Buildings; 43 Camp Sites; 19 Shrines; 11 Outcrops; 7 Ritual Sites; and More.
This is a bit overkill for me; I use a similar mod that only adds Player Homes (purchased or earned, such as the Archmage's Tower if you complete that storyarc) to the world map.
"Auto Un-Equip Arrows"http://skyrim.nexusmods.com/mods/10753A very menial, superficial mod for RP'ers. Automatically removes your arrow quiver off your back in 3rd person view when switching back to something melee - would otherwise create an ugly clash of items strapped to your back, especially if you use 2H Sword/Axe/Mace.
"Arrowsmith v3b - Simple Fletching"http://skyrim.nexusmods.com/mods/1178Allows for crafting of arrows via blacksmithing. Great for hard to find, powerful arrows used in endgame. Still has reasonable material/ore requirements, adds new inventory items to game to help in process. Use chicken feathers and quill pens to create fletchings for your arrows!
"Stones of Barenziah"http://skyrim.nexusmods.com/mods/9385For pickpockets and thieves - possibly considered cheating, adds world map markers for those $%^&@! stones once you find your first one *and* get it appraised at the Thieve's Guild. The completed crown greatly increases the spawn rate of precious gems for crafting in chests, urns, and dead monsters.
"Rich Merchants / Dynamic Merchants"http://skyrim.nexusmods.com/mods/769 or
http://skyrim.nexusmods.com/mods/8502Sick and tired of farming shit for gold early in the game, only to learn that most merchants are bankrupt after about 300 gold of transactions? This fixes that. "Rich Merchants" just ups the amount of default gold.. quick and easy. "Dynamic Merchants" keeps merchant gold from resetting, so a merchant is as rich as how much stuff you bought from him, and conversely can use that gold to buy back your unwanted items.
(Nexus Only)"Skyrim Character Manager V2"http://skyrim.nexusmods.com/mods/2119New players may not realize this, but Skyrim's default save system is only tailored for running one campaign/character at a time. Save game files are not self-nameable, and only mark down your location and a timestamp. If you're trying to play several characters at once, this grows confusing, and its possible to lose an entire character's career if you make a mistake. Quicksaves and Autosaves also indiscriminately overwrite each other regardless of character being played, adding another risk of losing hours of gameplay. Opening this mod (its a separate mini application) makes a full backup of ALL saves of your current character to a separate custom folder. You can then name a new character in this mod and switch to it, and then load up a clean instance of Skyrim with all your autosaves, quicksaves, etc. kept isolated. Do this any time you want to create a new character. The app does not need to be run every time you play Skyrim, only when you want to switch characters or create a new one.
"A Quality World Map - With Roads"http://skyrim.nexusmods.com/mods/4929Highly Recommended. Might be considered cheating depending on how hardcore you are with your exploration. Works as an amazing HD replacement of the default world map, but also comes with additional versions with major roads and even (optionally) all minor roads/trails filled in immaculately, as if Bethesda made it themselves.
"Cooking Recipes Pack"http://skyrim.nexusmods.com/mods/1368Cooking isn't used much after Level 10-15, especially if you're doing Alchemy. This mod helps keep cooking relevant with tons of new food ingredients and recipes to buff new/varying combinations of stats/abilities overlooked by the vanilla release. Also solves the problem of pure magic users having next to no beneficial food buffs available to them at all.
"Glowing Ore Veins 300"http://skyrim.nexusmods.com/mods/193MANDATORY for all blacksmithers or anyone trying to sell ore for gold early in the game. Makes the marble patterns indicative of raw ore in caves and out in the world wilderness actually visible without breaking immersion (at least not for me).
"Improved Skill Books"http://skyrim.nexusmods.com/mods/9960Puts books directly into your inventory upon pickup rather than reading. Also appends name of the book upon approach with whether or not it is a Skill-Up book, and of which skill.
"Ultimate Follower Overhaul"http://skyrim.nexusmods.com/mods/14037Improved follower AI and inventory management (they actually wear the stuff you give them and know when to switch from ranged to melee if you've given them the gear), allows for up to 15 followers, let's followers mount their own horses, and dozens of other features. Also adds new conversation options when talking to followers to better control their aggro/priority in healing and attacking. Highly advised you actually read the mod page well for everything this mod does and how to navigate it.
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Coming Soon: Highly recommended environmental / immersion mods that enhance the existing game.