I asked this question, because I remember when I tried to run this game on Windows 7 a year ago or so and failed, if I remember correctly I was getting a black screen.
After completing Fallout: New Vegas earlier this year I felt an enormous urge to sink in the post-apocalyptic world again and decided to finally finish Fallout 1 & 2 (I have played both when they came out, but I was a different type of gamer back then and never finished either), however remembering the issues I faced previously I figured I'd ask first. Shadur's answer is OK if you don't own the games already, but I do, so I didn't want to re-purchase them. Instead I decided to give it a try on my own again and here's what I came up with...
First of all, for some people your antivirus software may complain about trojans and other nasty things in some executables on the DVD. I am using NOD32, but I saw other users report the same issue with different antivirus solutions. My copy of the game is legitimate and bought in a game shop from official distribution, so this is a false alarm (I hope! :P). It's best if you disable the antivirus for the duration of game installation.
Run the installer with administrative privileges. So instead of letting the autorun program deal with the installation, close it and start the installer manually by running it as administrator. Then proceed normally.
The latest official version of the game is 1.2, which sadly still has bugs. That is why it's great to have a vibrant community and Fallout always had that (and still has)! Modders have released several unofficial patches to the game, so go to No Mutants Allowed and grab the newest one you can find that suits your version of the game (usually 1.3.4). Just unpack it to the folder where you have the game installed overwriting any files it asks you about.
Because we're on a modern machine, it would be good if the game could look a little bit better. You should get something called Sfall with the community patch. It's a modification that will allow you to run Fallout through DirectX 9 instead of DirectDraw, letting you to change the resolution and scale the graphics through the use of some scaling filters (known from console emulators, such as ZSNES). This one didn't work well for me, when I tried to run the game in a higher resolution in fullscreen it would crash. Higher resolutions worked in windowed mode, but then the performance was sluggish. There is another option, however, namely the Fallout 1 High Resolution Patch 2.2b. It adds a new set of options on the main game screen and you'll be able to change the resolution there. In order to make it work with Sfall properly, I'd recommend setting Mode=0 in ddraw.ini, so that Sfall doesn't try to use dx9, HRP will change the resolution and color depth instead. Good settings are probably 1024x768@16bit, higher resolutions may make the dialog windows too small.
When you run the game, it's probably good to do that with administrative privileges, just in case.
And that's it! The game runs very well on my Windows 7 x64, the performance is excellent and I didn't encounter a single crash yet. I am playing at 1024x768 resolution and have set HRP to resize the ingame movies while maintaining the aspect ratio. You should really review ddraw.ini (for Sfall) and f1_res.ini (for the High Resolution Patch) and modify the settings to your liking.