Download and install DVD ripping software. Two of the most popular ones are Handbrake and Mac the Ripper. Handbrake works for both PC and Mac, and Mac the Ripper works only for Mac. Both programs are free.
Put your DVD into the DVD drive of your computer. On either program, the DVD will show up as the source file. Adjust the settings of the output format according to your needs. You will have place to select aspect ratio, quality and audio settings. Set these as close as you can to the original DVD. You can also set the program to match these settings to the original DVD files. You will want to rip the DVD as an MPEG-4.
Import the MPEG-4 file to your video editing software. If you don't have software, some good ones are Final Cut, Adobe Premier and Avid. There are also free ones such as iMovie for Mac and Windows Movie Maker for PC. Edit the video to your liking. These programs have similar interfaces. There will be a time line or sequence at the bottom of the screen for editing the video, as well as a window to see your edits.
Export the edited video as an MPEG-2 file. Click "File>Export" and choose MPEG-2 as your output format.
Remove the original DVD from the DVD drive and insert and blank DVD-R.
Import the MPEG-2 into a DVD authoring software, such as DVD Studio Pro, iDVD or Adobe Encore. These programs have similar interfaces. There will be a window to drag the imported MPEG-2 file to burn to DVD.
Click "File>Burn to DVD" in your DVD software. Wait for the DVD to finish burning and you're done.