http://sc2.sourceforge.net/
This is a modern port of the Star Control 2 game worked on (at least initially) by part of the original team. Its not at a 1.0 release but is very stable, and as far as I can tell, fully playable.
The day I found this was one of the happiest gaming days I've ever had. Check it out!

Ovine by Design made a fabulous remake of "Imogen", which was a whimsical 2D platform/puzzler on the BBC Micro. They also remade "Cholo" which was an early (post-Elite) wireframe 3D game which I had never completed in its original incarnation, so that was fun to play through. Cholo was ported to many systems, so is probably familiar to more people.
And slightly off-topic, but... "Sinistar Unleashed" is, bar none, the best 3D recreation of a classic 2D arcade game that I've ever encountered. Absolutely brilliant. I'd love to see it re-released here at GOG.

Ultima V: Lazarus is fan remake of CRPG Classic Ultima V: Warriors of Destiny made by Team Lazarus using Dungeon Siege Engine.
It requires original Dungeon Siege game or it's expansion Legends of Aranna (includes the original game) to run. You may be able to find it cheap in bargain bin or get used version from someone. I strongly recommend the use of fan made bug fix collection as it fixes number of bugs left in final 1.20 version.
If you like UVL, you may want to check The Ultima 6 Project, a remake of Ultima 6: The False Prophet being developed by Team Archon also for Dungeon Siege Engine. The project is at it's fifth milestone having most of the surface world implemented. You are free to test the M5 release if you like; just keep in mind its basically public alpha/beta, not a complete game.
UVL and U6P both utilize Project Britannia, framework and world map they jointly developed for creating Ultima like games using Dungeon Siege engine.
You can find links to other Ultima and other fan projects and to tons of freeware games from my site.
edit: forgot there's no signatures here :p added link to my site heh

Two words: Black Mesa.
That's going to be awesome. :)

I've played another Ovine's remake, namely Total Eclipse. It's rather complex and difficult 3-d labirynth game, where you need to find your way to the central chamber of a pyramid before an eclipse occurs, destroying all life on earth.

http://chaosgroove.shiftcreate.com/
The best game of all time given a wonderful 21st century update by Author, Richard Phipps. If you loved Chaos on the Atari ST you're gonna flip-out when you play this!

Super Street Fighter 2 Turbo HD Remix ( that's a mouthful!) Is a pretty awesome remake, remix of street fighter 2 games. the fresh graphics and music are amazing! The online play is great too! I have came across some real street fighter fans who have some serious skill.

Speaking of Ultima remakes there is always Exult for Ultima 7

Wow, thanks for the heads-up. I played Driller and Total Eclipse on the C64, and not only did I not do very well, I don't think I actually understood the game either... I look forward to giving the remakes a bash.
... I wonder if anyone will ever remake Puzzle Panic.

Remakes I've really enjoyed include:
Space HoRSE - a remake of M.U.L.E, a much-loved game from the 80s
HeroQuest - a remake of the DOS game, and not a bad rendition of the board game either.
Xenon 2000, which is more of a tribute to Xenon 2 than a remake, but made by the Bitmap Brothers themselves
Scorched3D - a remake of Scorched Earth. Who didn't love Scorched Earth?
ROM CHECK FAIL - a remake of about a dozen games at once. Small and silly, but I love it.
(Edit: ROM CHECK FAIL)

the only remakes by fans, that ive played are the kings quest ones.
i would like to see toonstruck and dungeon master remade though (i know dungeon master java exists, but it aint a remake, its new levels etc.)

Here's a cool fan demake of the popular Team of Fortress game. The game is called Gang Garrison.

I thought Prince Of Persia Classic on the xbox live arcade was an excellent remake. Basically the same game as the original with sands of time style graphics and a modified and much harder fighting mechanic

Coming Soon!
Well, "late 2009."

Yeah, I saw some videos of that game on Youtube and I must admit it was amazingly well done.
Too bad they didn't release it for the PC :(