Hi and welcome!
Your friend is correct to point out the difference. Here are the basics.
Optical image stabilization means that either the lens or the sensor have moving elements that are used to compensate for some of the "camera shake" that users can generate. This shake is more visible in low light (when you are exposing the image longer, thus increasing the risk of moving ever so slightly) or when using a somewhat long zoom (every tiny movement is magnified by a long zoom). Image stabilization is not a miracle tool, but can help you take pictures in tricky situations.
Digital image stabilization is, honestly, a mrketing gimmick. All it does, usually, is increase the ISO value (in other words, the gain of your sensor) in order to be able to reduce the exposure time and limit the risk of moving. Doing so, however, increases the level of digital noise in your image. And with compact cameras, the sensors are usually small and VERY sensitive to noise.
Increasing the ISO of he camera can be useful and/or necessary in some situations, but calling it image stabilization is excessive and misleading. That's not to say a camera without "real" image stabilization is bad, but the term "digital image stabilization" is more misleading than it is accurate.
Some helpful readings:
High ISO:
http://www.digicamhelp.com/camera-features/camera-modes/high-sensitivity-mode.phpImage stabilization :
http://www.digicamhelp.com/camera-features/camera-parts/image-stabilization.php