Place your digital camera or camcorder on a flat surface or tripod, directly in front of the surface onto which you'll be projecting your film. Set up the camera so the entire film frame will be visible to it. Remember that this digital copy is what is going to be around forever; you don't want to incorrectly frame your picture at this point and lose any visual information. Be sure to white balance your camera and make sure that what is recording is in focus.
Project your 8mm home movie onto your projector screen or flat, white wall. Make sure your projector is displaying everything as focused and clear as is possible. Run your film through all the way a few times, especially if you haven't done so in a while, to make sure everything is going to run as intended.
Use your camera to record your 8mm film. You may want to record a few tests to make sure your camera is getting the results you want. Make sure it doesn't do anything unexpected in the middle of the recording process (like autofocusing on another object), completely ruining your results.
Transfer your recently recorded video to your computer. Once you are satisfied with the results, you can transfer your video from digital tape (or just copy a digital file, depending on the type of camera) to the hard drive on your computer. From there, you can do whatever you want with it, like burning it directly to a DVD, adding music and other audio, or adjusting the picture settings or color.