All digital zoom does is basically crop the image you take, this means you get a greatly reduced megapixel count, which equals a loss in quality, instead of using digital zoom, it's better to take the image full size and simply crop the image yourself on a computer when you get home, if you want to do so.
Optical zoom uses the lenses in the camera to increase the focal length, basically this does what it says on the tin, it zooms in, just like binoculars would do (random example), and the resultant picture has the same megapixel count as it would if you had not zoomed in, therefore there is no loss in quality.
Do note though, that when you optically zoom in, camera shake will have a greater effect on the image because you're focusing on a smaller area, and could blur the picture more noticeably (The same with digital zoom, but because of the drop in quality it would be less obvious, though to the same extent). That's why you should go with a camera that has optical image stabilisation (Usually referred to as OIS), it will greatly reduce image blur.
EDIT to Shorty below: If you get a good quality camera (High end compact, bridge camera, or DSLR) then the barrel distortion will be very minimal, if any at all.