If you have the problem that the screen is going black after removing the ddraw.dll and you can "erase" the black screen with the cursor, try this patch:
sfall
It is the "Fallout 1 v1.2 US" that you need, according to the readme it won't work with the 1.2 UK version, so you'll have to upgrade, I use the 1.3.3 myself without problems, but I'll add a link for the 1.3.4 as well. (I haven't tried that version myself though)
Patches:
1.3.3
1.3.4
Note on above patches, they won't work with your old saves, so you'll have to start from the beginning.
In the sfall patch you'll get a ddraw.ini, open it and change mode to 4 under the "graphics" section.
This will cause corruption of the in-game movies (just like stated in the .ini), but IMHO it's better than the screen going black without the original ddraw.dll or having the sluggish gameplay you get with it.
I thought this topic might be useful to have on its own to prevent people from asking about it more than once and since this info so far has been "hidden" in other threads.
For Vista users the above won't work as described, check this to find an easy step by step guide:
Vista fix
Thx to "404notfound" and "matthewmw64" for finding the way to do it and for compiling the step by step guide for Vista.
Edit:
Ooops, I forgot to add that this concerns Fallout and not Fallout 2, so if some admin could add "in Fallout" or "in Fallout 1" after "Screen going black" in the topic I'd be grateful.

I've followed your instructions to the letter and I keep getting this error message: http://i80.photobucket.com/albums/j184/matthew64/FalloutFailcopy.gif
Any ideas?

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I'd say (since I remembered getting a problem with some ddraw.dll that I DL'ed, got it now again to see what happened and I got that exact same message) that you didn't manage to overwrite the existing, or somehow got the wrong one.
Maybe you'll need to be in admin mode to do it? (I don't use Vista so I can't try it myself unfortunately)
Check the ddraw.dll that you have in the FO folder, mine says "1.18.4.0" under the name, checking the version under properties it says "1.18e".

Nope, still no luck :(
Methinks this fix unfortunately won't work on Vista.

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Hmmm, that'd be typical. :(
I tried some things before to check if I could find anything that changed it working, removing the .dll, removing the .ini and removing the .dll that came with high resolution pack. (the latter did nothing, the other two made the blackness come back, but game still worked)
It's strange that you get the exact same problem that I get if I replace the .dll that came with the patch with some ddraw.dll that I downloaded from somewhere. (which is why I thought you might've gotten the wrong one)
Looking at the readme that came with it, I saw this:
"you'll need to patch it into the US version before using this mod."
Have you upgraded your game beyond the one you got from gog?
If not, that might be the cause, I use 1.3.3 myself, which is available here:
http://www.teamx.ru/eng/files/f1patch.shtml
Or if you prefer you can go to NMA and get the 1.3.4, the reason I didn't is that you'll have to wait to DL unless you're a member at atomicgamer, at teamx's place you can DL it instantly, here's the 1.3.4 at NMA though:
http://www.nma-fallout.com/forum/dload.php?action=file&file_id=1075

Still no luck. I hadn't upgraded beyond 1.2, but i did and that also did nothing.

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That's typical, I'm afraid I've run out of ideas myself, did find this on NMA's forum, but it's for XP, but since it's HW related it might be worth a shot:
"Disable AGP Fast Writes in the System bios:
this is usually under chip set or PCI settings.
Disable AGP Write Combining:
For the Latest Nvidia Forceware Drivers this is under
display properties -> settings tab -> advanced -> troubleshooting
and may only show up under certain hardware / driver combinations "
Found another solution, but this is meant for XP as well, still might be worth a shot though, get the FO1 screen refresh here:
http://www.labo-asso.com/php/outils/fallout.php
According to people on NMA it's just to run the program before starting FO and leave it running in the background, to quit it when you're done playing FO press "ok". (it's French)
I did of course not go through the entire NMA forum, all I did was a search with the term black AND screen, maybe black AND vista could give more results, so if none of the above works maybe you could try some different searches on NMA's forum for a possible solution. (unless the NMA'ers that we have here knows of something I've missed)
Hope you'll find some way to fix it, if/when you do, please post it here so everyone else with the problem and vista can see how you did it.

Oh, sorry, I forgot about that, I'll add that above too, the 1.3.3 patch doesn't work with the old savegames. (doubt the 1.3.4 does as well)
But you can keep those, just copy them elsewhere and move them back after reinstallation. (since you'll be back to the same version again)

Well, reinstalled everything and I'm still getting the same problem. Beginning to think this is almost certainly a Vista issue. Might be worth asking about on the NMA forums.

I just found out about this fix a few hours ago myself, but try renaming Falloutw.exe to something else and then running it. Timeslip's sfall for Fallout 1 won't work under Vista unless you rename the executable, for whatever strange reason.

The Hell? That sort of works, but it seems to be making the Res patch not work properly. Will need to investigate.

What do you mean when you say that the resolution patch doesn't work? Do you mean the program that applies the patch? It would make sense for it not to work, in that case, since it's looking for a file specifically named Falloutw.exe.
What I did was I used the res patch on Falloutw.exe, and then I renamed it to make sfall work. It's been working fine for me that way so far on my Vista laptop.

May I now present my Guide to getting the Res patch to work on Vista.
With special thanks to Rainstorm and 404notfound.
1. Install Fallout as normal.
2. Download the Resolution Patch (http://nma-fallout.com/forum/dload.php?action=download&file_id=739)
3. Download the FO1 Engine Fix (http://timeslip.chorrol.com/sfall.html) Its the one for XP/Vista.
4. Install the Resolution Patch following the instructions and then the same for the Engine Fix
5. DO NOT change the graphics part of the DDraw.ini file. You will not be able to do anything at the main screen.
6. Rename the falloutw.exe to something else. (Update shortcuts if necessary)
7. Enjoy!
That should work. Please note that I've not had much time to play after figuring this out so the black out problem could still reoccur. Let me know of it does.

Its kinda hard to describe what happened, but it was because i had Mode=4 in the ddraw.ini file.