Recovering such DVDs is not a simple task: if you get really lucky, its a piece of cake, but normally it ranges between very difficult and extremely difficult, even for geeks. If you aren't a geek, prepare to become one.
The first thing you need to do is verify what type of DVD these actually are: DVD+R or DVD+RW. The RW symbol does not necessarily mean they are RW but plain +R that is compatible with RW drives (some blank disc mfrs use strange logic in their markings). Plain +R is a little easier to recover videos from, while +RW can be much trickier. Try loading the DVDs in your computer, then look at how Windows identifies them under "E: Optical Drive." It should display as "E: DVD+R" or "E: DVD+RW." Plain +R is capable of the easiest and best fix: finalizing on the same brand of recorder. But +RW media doesn't have a finalizing ability, so they must be tediously recovered using complex PC software.
It would also be helpful if you can find out what brand of DVD recorder made the discs in the first place. Some brands were more amenable to DVD recovery than others, and some are still being sold today that use the same finalization code they used ten years ago. So ask your Aunt to think back and try to remember the recorder brand she had. If it was Philips or Magnavox, these brands were the same units, and are still sold today. The easiest fix for many +R (but not +RW) DVDs made on these brands would be go to a store that has a display model and offer them $20 to let you finalize a few discs ("finalizing" makes a +R compatible with devices other than the recorder brand). Or, buy a Magnavox from Walmart, the cheapest one is about $99 when on sale.
(Philips and Magnavox use a somewhat peculiar method of recording that makes recovery of unfinalized DVDs more complicated than usual: I strongly recommend use of a matching new recorder instead of trying different computer software solutions. The Toshiba brand was absorbed by Philips/Magnavox in 2006, so Toshiba could also be an option depending how old the original recorder was. Any of these three brands can usually finalize each others +R discs, or play funky +RW discs out to a second recorder or PC to make new copies: see DigaDo's post below for exceptions to this.)
If your Aunt's old recorder was a brand that used the more traditional coding method (such as Pioneer, Sony, Panasonic, LG, pre-2006 Toshiba, etc), more possibilities open in terms of computer software that might be able to pull the videos from these discs. Recovering videos from peculiar DVDs is kind of a Windows-specific task nowadays: not much software is available for Mac users. So if you don't have a Windows PC available, most suggestions you find on AV forums won't apply. One typical discussion can be found at
this link , which shows how involved and tricky DVD recovery can be.
If only a handful of discs are involved, another option is to download the 30-day free trial version of
Cyberlink PowerDVD to a Windows PC. This media player has a unique ability to read and play videos from most unfinalized or damaged or eccentric DVDs that Windows itself won't even recognize in the drive tray. However, PowerDVD cannot digitally capture what it is playing to a new file on your computer. If your PC has an analog video output, you can play the videos from PowerDVD into
another computer or DVD recorder to make normal DVD copies of the material. If your PC only has HDMI video output, you might need an HDMI>Analog adapter ($30-$60). Some versions of Nero software have similar capabilities, and depending on the original recorder Nero can sometimes repair/finalize problem DVDs by itself.
