There are a few different ways you can copy your old VHS tapes to a DVD and I'll explain them all.
VHS to DVD Conversion Services
There are many companies out there that will actually copy your old VHS tapes for you. They charge anywhere from $10-$30 per tape, but if you have a lot of tapes to backup, this could become a very expensive method.
VHS/DVD Combination recorder
There are many VHS/DVD combo systems out there that play both formats, and a few of them are actually capable of recording to DVD. If you have a combo unit you may have a DVD recorder right under your nose and never even knew it! If you happen to have one of these units, follow the instructions found in the user manual and easily backup your VHS videos to DVD.
Digital Video Camera
If you have a digital video camera, most of them are capable of converting analogue video to digital quite easily. Hook your VCR up to your digital camera using composite cables, press play on your VCR and the record button on your digital video camera. Once you've recorded the entire tape to the digital video camera, you can copy the data to your pc and burn the video on to a DVD.
To copy your VHS videos to your computer, you need either an analogue video input on your video card or an analogue to digital video converter. You can get an analogue to digital video converter that plugs into a USB input on your computer. (An example of one is the Plextor Convertx External Digital Video Converter USB 2.0 and cost about $65). Once you have copied the video footage onto your computer, you can use software like Windows Movie Maker to edit and convert your video into whatever format you want prior to burning it to DVD. Using the computer is my favorite way to convert VHS to DVD since you can update your videos with subtitles, menus, transitions and other cool things that actually make your old videos even more fun to watch.
External DVD Burners
If you don't have a computer or the time to mess around with learning how to copy or edit video, an External DVD Burner might be the right option for you. You can buy the hardware for around $100, and it just plugs straight into your VCR. Press record on the VCR, and record on the DVD burner and by the time the VCR is done playing, you have a nice quick copy of your VHS tape.