The basic difference between analog and digital sound is the way it was recorded and is stored.
According to wikipedia, analog recording
is one where a property or characteristic of a physical recording medium is made to vary in a manner analogous to the variations in air pressure of the original sound
Ughh…Seems slightly complicated? Well, analog sound is basically a cast of real sound caught in specific moment. It is stored by replicating air pressure of original sound by magnetizing a tape or by cutting grooves on a vinyl disc (sweet times…). When it needs to be reproduced, magnetic tape moves (or needle runs along the grooves) causing fluctuations in electric signal which is then amplified through your expensive (or not) dynamics. This whole process is continuous which is why many people associate analog sound with something smooth or even “warm”.
On the other hand, digital sound is nothing more than a sequence of ones and zeros stored at one’s disc or hard drive. This information represents sound fluctuations in specific (very small, as you can imagine) time intervals. So, unlike analog sound, which is continuing flow of signal, digital sound moves in steps. Very, very tiny, but nonetheless. Reproducing it is a little more tricky, because your dynamics expect continuous flow of signal, therefore, such thing as digital-to-analog converter should be used. Digital sound can be reproduced or copied without any loss of quality unlike analog records which wear down with time and use.
There is never-ending holy war between which one is the best. Digital sound is “cleaner” from outside noise and is more reliable when it comes to storing, copying and reproducing sound. But there is definitely something magical about smoothly flowing analog sound. We will not take sides in this battle and advise you to enjoy good music in all possible manners.