Your Current TV - Analog Signal
As analog signals weaken over a distance, TV receivers have difficulty completing picture, just as you may be able to pick out pieces of a co-worker's conversation across the room. With the conversation you may not be able to hear each word, but you may be able to understand the meaning of what is being said. The same is true with analog television, as the signal gets weaker the picture may not be perfect or exhibit "snow", but you can still make the picture out and understand what is being shown.
Digital Television - Digital Signal
Digital signals do weaken over a distance, but unlike analog signals the complete picture is impossible to put together unless all the information transmitted is received by a digital TV tuner. Therefore the picture one sees on a digital TV is EXACTLY the same picture that was transmitted - free of snow and ghosting.
Digital signals can also be compressed, whereas analog signals cannot. Therefore digital TV can put more information into the same channel used by analog TV. This means that digital TV pictures can be sharper, can include 5.1 channels of digital audio, and even data used for interactive television.