If you have a computer running Windows Vista, then your computer is likely new enough to have a camera built into the screen. If your computer does not have a built-in webcam, then you can purchase a webcam. Most computers, built after 2005, have webcams built-in. If your computer does not have a webcam, then I suggest either a Logitech or Microsoft brand camera. You can purchase them from most stores that sell electronic or from Amazon.com (http://www.amazon.com/s/ref=nb_sb_noss_1?url=search-alias%3Daps&field-keywords=webcam) and they should cost under $50.
Then, you'll want to go to http://www.skype.com/en/download-skype/skype-for-windows/downloading/ to download a software program called Skype. Skype may come pre-installed on your computer or may be included with your webcam, but it is easy to download and Skype is completely free.
There is, also, a pay version of Skype, if you want to do video calls with more than one person, but in your case, you want to see your great-grandchild, and that service is completely free of charge. They, also, make Skype cameras that will attach to your Television set, if you wish to have an even larger screen to see your new grandchild. These cameras do cost more, but the advantage is, you don't need a separate computer, and the image is a lot larger, because it appears on your television screen.
Skype is a very simple and easy to use computer program. All you need to set it up is an e-mail address. Most cameras, even if they're external, are very simple to install. If you're prefer, you can even get someone, like BestBuy's Geek Squad to set it up for you, but you should be able to do it yourself, with very little effort. Your grandchild will need Skype, as well.
The alternative to Skype, would be a Apple FaceTime. Apple FaceTime will not run on Windows Vista, but it does run on an iPhone, an iPad, an iPod Touch and an Apple Mac. If your grandchildren have Apple products, this may be their preferred way to video chat. Apple FaceTime is free, with these devices (except for the Mac, where its very affordable). The iPod Touch ($200-$300), would be the least expensive Apple device, that runs FaceTime, followed by the iPad Mini ($330). Both of these devices are incredibly easy to use. Apple products, can also run Skype, and all of these devices, include a camera. Skype is free on these devices.