Your Current TV - Analog Signal
The word analog comes from "analogous" - that is to say an analog signal is analogous to the original signal such as a voice or video. A analog signal is often represented by a "waveform", as seen in the image above. For example if a voice is louder, the height of the waves are higher and if someone speaks in a higher pitch, the waves are closer together. The same applies to the brightness and color of a video 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
A digital signal represents the original signal by a series of "1s" and "0s". A "sample" of the original signal is taken at specific interval. For example a CD's sample rate is 44 KHz or 44,000 times per second.
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.