A very short explanation:
DNG, also called RAW-files are raw, unprocessed data taken directly from the camera sensor. To view them as a photo, you need to process them in a software called RAW-converter. Popular ones are Lightroom, Camera Raw, DxO and several others. Here you can adjust the photos in every way you want - colour, sharpness, contrast etc.
JPG's on the other hand are RAW-files that are processed in the camera. The engineers at the camera manufacturer have decided what is the best way to adjust the photos, they are sharpened, colours are saturated, contrast are increased, distortion corrected etc.
The good thing with DNG files is that the photographer has much more control of the finished result. The files can take far more adjustment than JPG-files, and they have larger dynamic range.
JPG's are great for those who don't care so much about post-processing, but think that the photos look fine as they are from the camera - which they often do. JPG's are not necessarily "worse" than DNG's, but you have less flexibilty to adjust them.