This is a list of mods that alter the game's world or mechanics in some way. Obviously there are far too many mods out there for me to list them all, but these are all mods which I personally recommend and which I have judged to be of good quality.
Read this article if you're not sure how to install mods, and please if any links are broken.
Trancemaster's Morrowind Rebirth is a compilation of content and gameplay mods that can be used as a base or as an alternative to the mods listed on this page. I have not used it personally, but I understand it changes the game fairly drastically.
All of Bethesda's official plugins, plus fan-made patches.
Install before any relevant graphical replacers. Manual installation is a little confusing. The easiest way is to install all the assets from the "01 Original Files" folder, then install (and overwrite) "Photodragons.NIF" with the one from the Meshes folder, then use whichever plugins you want from the "00 Individual Patches" folder.
Skip LeFemm Armor if using the version from Moranar's Better Morrowind Armors.
You should of course already have the most important bugfix of all, the Morrowind Patch Project by Thepal and Quorn.
Fixes around 120 floating or otherwise poorly-placed objects.
Makes killing Ash Vampires weaken Dagoth Ur correctly, as Bethesda originally intended.
Plays a previously-unused sound when the player is affected by the Sound spell effect.
Tidies and fixes syntax errors in almost 1200 scripts, for greater stability.
Removes the scroll interface from plain paper, reduces the weight from 1.00 to 0.05, and reduces the value from 3 to 1.
Note that the Morrowind Patch Project already reduces the weight to 0.20, and corrects the lowercase item name.
Fixes some bugs with the two Guar followers.
Drastically changes many elements of the game.
Includes flee AI, a magic system overhaul, dialogue and faction-relation enhancements, and rebalancing of numerous items / characters / game settings.
These changes are divided across 8 plugins, to be enabled as desired:
Balance - Character: Balances experience gain, races and birthsigns.
Balance - Game Settings: Adds flee AI and tweaks a number of settings, e.g. giving NPCs more magicka, making diseases more likely, and increasing the difficulty of lockpicking.
Balance - Items: Balances item costs and stats, improves consistency between items, and adds enchantments to several overlooked artifacts.
Balance - Magic Effects: Changes the costs of most spell effects to make them more balanced.
Balance - NPC Spellcasting: Adds new spells that will be picked up by most NPCs, designed to make them less stupid spellcasters.
Balance - Spells: Tweaks most of the game's spells to make them much more useful, and updates the casting costs based on Balance - Magic Effects (required).
Gameplay - Dialogue: Fixes a few dialogue bugs, makes some generic dialogue less common, and makes NPCs' behaviour more consistent with their dialogue, e.g. they might run away from you if you are a criminal or diseased.
Gameplay - Faction: Changes guilds to be more stringent in their requirements, and makes most guild members refuse to serve non-members.
An all-in-one version is available here, although this apparently requires cleaning.
A noteable alternative is BTB's Game Improvements, which expands on Wakim's Game Improvements and also takes inspiration from HotFusion's Economy Adjuster. I have not used this personally, but I understand that some of its changes are very drastic.
Changes the governing attributes of several skills to ones that make more sense.
Makes Magicka regenerate over time based on willpower.
For alternatives, see Gluby's excellent Comprehensive Guide to Magicka and Spell Effect Mods.
Changes a number of game setting to make it much harder to accumulate gold.
Granted, I made this myself, but personally I never play without it.
Also consider Piratelord's Trade Enhancements, which overhauls trading and looting and reprices virtually all items in the game. Traders no longer have infinite stock, and will restock over time with random - and sometimes rare - items.
Switches around the open-top cantons of Molag Mar and the closed-top cantons of Vivec.
Based on Szazmyrr3's Vivec Replacement. Also includes a small house in the Foreign Quarter Plaza.
Will conflict with most mods affecting Vivec's exterior.
Allows fast-travel from any boat port to any other boat port.
Allows fast-travel from any silt strider port to any other silt strider port.
Adds a hot-air balloon to Gnisis, Ald Velothi and Ghostgate, and fast travel between them.
Allows real-time scenic silt strider travel.
Allows real-time scenic boat travel.
Allows real-time scenic gondola travel.
Makes houses and shops get locked at night.
Adds doors and walls to a number of houses and shops to prevent trespassing into private areas.
Part of Antares' Little Mods collection.
A story-based dungeon at a haunted inn just south-east of Khuul.
There is an optional plugin available for download separately that adds a new NPC to Khuul with some extra backstory and directions (recommended).
Triggers disturbing happenings around Vvardenfell as the main quest progresses.
Makes leading NPCs less of a chore by increasing the speed of all followers.
Adds a hermit west of Seyda Neen with 4 associated quests.
Grants the player certain privileges at higher ranks in guilds.
Adds over 400 new dialogue entries to make dialogue less generic.
Reduces the prevalence of the generic Solstheim rumour.
Restores some dialogue written by Bethesda that is not used.
Removes lore-related topics from "no lore" characters, and removes generic responses once they stop being useful.
Based on ManaUser's NoLore.
Prevents non-Telvanni players receiving services in Sadrith Mora unless they have Hospitality Papers in their inventory.
Makes most merchants refuse to trade in Dwemer artifacts.
Ash Vampires no longer attack the player outright, allowing for some Bethesda-written dialogue to be seen.
Adds hundreds of new guards in place of the old generic ones.
Adds some new animated NPCs around Vvardenfell, with 15 new animations.
Adds 37 more animated NPCs, with 9 new animations.
Automatically and visually harvests plants upon activating them.
If using Piratelord's Expanded Sounds, you also need this patch by philologos.
Adds Bloatspores to the wilderness and caverns of Morrowind (made by Bethesda but never added to the game).
Changes potion names so that they are sorted in a more logical order.
Removes the annoying "item does not have enough charge" message (the sound effect is sufficient).
Adds new scripted spells and new NPCs that sell them.
Adds some new crossbows and tweaks the stats of various ranged weapons.
Adds over 130 new enchanted weapons to levelled lists.
Includes a patch for Bravo1's Marksman Enhanced.
Includes quorn's Steel Broadsword Fix, thus removing the need for that mod.
Adds 12 new artifacts to Morrowind.
Adds 241 amulets and 130 rings to random loot.
A collection of small mods, mostly gameplay tweaks.
I recommend all of them, except perhaps the following:
Mixed Guards 2: NaxoX9's NX9 Guards Complete does this, and more.
Mortal Guards: Personally I want guards to respawn.
Show Ammo Dmg - NoExp: Obviously don't use this if you have the expansions.
Stolen Goods Fix - NoExp: Obviously don't use this if you have the expansions.
A collection of small mods, mostly graphical and balance tweaks.
Some of these mods were already covered by the "Graphical Mods" page of the guide. I recommend all of them, except perhaps the following:
Creeper and Mudcrab NoTrade: Personally I like Creeper and the Mudcrab Merchant being able to trade.
GMSTs: These are mostly economy tweaks covered by other mods.
Negative Lights Remover: Not needed unless using MGE XE's per-pixel lighting.
Scroll Fix: Not needed if using Vurt's Hi-Res Scrolls.
Smaller Potions: Not needed if using the half-size potions from Better Meshes (although these are 60% of the original size, which may be preferable).
I was very happy to see that PeterBitt had made Gondolier Destinations based on an idea I posted :)
Gives keys more descriptive names.
Makes the days twice as long.
Simply makes the game start at night, giving it a very different feel.