Recommended Mods

Engine

Morrowind Code Patch (MCP) - Hrnchamd
Incompatible with OpenMW
Modifies Morrowind's exe to fix bugs and optimize things. This allows you to scale up the map to show TR's landmass, but still being a fixed size.

Morrowind Graphics Extender XE (MGE XE) - Hrnchamd
Incompatible with OpenMW
Allows Morrowind to render distant views, scenery shadows, high quality shaders, UI scaling and other features. Works with the original engine, so can be combined with the Morrowind Code Patch but will not work with OpenMW.

OpenMW - OpenMW team
Incompatible with MCP and MGE XE
An open source engine replacer for Morrowind, fixing many bugs with Morrowind's engine. OpenMW allows for better performance of Tamriel Rebuilt on modern hardware. It also allows the map to be as big as your mods need it to be, at a higher resolution than MCP.

Bug fixes

Patch for Purists - Half11
Patch for Purists supercedes the Morrowind Patch Project/Unofficial Morrowind Patch at this time by still being maintained and limiting itself strictly to bug fixes, with gameplay changes confined to optional plugins. Notably has implemented some bug fixes in coordination with TR and the PT mods.

Morrowind Optimization Patch - Remiros & Greatness7
This mod greatly improves outdoor (and to a degree indoor) performance in Morrowind, as well as fixing a lot of errors related to flora.

Project Atlas - Melchior Dahrk and the Project Atlas Team
An extension of Morrowind Optimization Patch which makes models use only one texture instead of many, improving performance. Install after Morrowind Optimization Patch.

Essentials

Delayed Dark Brotherhood attack - Kir
Based on original Avenger's Delayed DB Attack, this mod makes Dark Brotherhood assassins only attack player in his sleep after he reached some significant point in the main MW plot (and accordingly, decent level).

Project Tamriel

Skyrim: Home of the Nords - SHotN Team
Adds in a portion of Skyrim into the game: the Reach near Markarth.

Province: Cyrodiil - P:C Team
Adds in the island Stirk located west of Anvil.

Graphics

Near Vanilla Road Sign Replacer -  Atrayonis
A mod for the few people who want to keep graphics as close to vanilla as possible, yet still don't like the illegible road signs that shipped with Morrowind GotY.
Covers Tamriel Rebuilt lands as well as Vvardenfell.

Arukinn-Style Signposts for Tamriel Rebuilt - Red Furry Demon
Requires Arukinn’s Better Banner Signs and Signposts
This mod replaces all the road signs on the mainland with readable ones similar to those in Arukinn's Better Banners Signs and Signposts. The style isn't exactly the same (would you expect one Mer paint ALL the signs in Morrowind?), but I hope it's close enough. Texture size is 1024x512.

Grass

Aesthesia Groundcover - Rytelier
Requires MGE XE
Makes every landscape covered in rich grass, made with vanilla aesthetics in HD. Lands will feel fuller and more lively with minimal to no performance cost on average PC.

Remiros’ Groundcover - Remiros
Requires MGE XE
This mod adds high quality grass and groundcover to almost all regions in the game as well as all Tamriel Rebuilt areas. It also includes modified MGE XE shader files that add support for Vcol-based distortion for groundcover objects. This is absolutely required for this mod to function correctly.

Miscellaneous

Anthology Solstheim - Atrayonis
Incompatible with most Solstheim mods
This mod simply moves Solstheim to a lore friendly position, further away from the mainland and Baan Malur in particular.

Better Dialogue Font - Hrnchamd
This mod makes Morrowind's font clearer without changing the font size.

Better Daedric Font - hardek
“A high resolution replacement for the Daedric font used in scrolls.”

Graphic Herbalism - MWSE and OpenMW Edition - Stuporstar & Greatness7
Requires MWSE Nightly or OpenMW Nightly.
Changes the way you pick plants- instead of opening a container, you simply activate them to harvest their contents, with a visual change upon harvest.
Includes Tamriel Rebuilt flora.