The most popular program right now is DVD Shrink. It's free, and does a better job of compressing DVD movies than the paid programs. It both rips encoded DVD's and lets you "backup" either the movie-only or the whole DVD. It has a preview screen that shows video of the different files. Get it free fromhttp://www.dvdshrink.org/
Yes, Nero Burning ROM will burn the movie DVD's. All of the files created by DVD Shrink need to go into a folder titled VIDEO_TS (must be in all capitals). If you tell Nero you want to do a Video DVD it will create the VIDEO_TS folder for you and you can copy the files into it. Many DVD's also have AUDIO_TS folders, but they aren't necessary, and you don't need to put anything in them. Remember that DVD movies typically range from 4 to 9 GBytes in size. Recordable DVD's are typically 4.472 GBytes, so the files need to be compressed further. This is why many people copy only the movie. This also makes it so that the movie will start automatically without all of the menu stuff and previews. If you do want the whole DVD, DVD Shrink will let you compress different parts of the DVD different amounts. Without a whole lot of work using other programs, you can only choose the whole DVD or Movie Only (not the movie and one of the special features).