An analog system is a mathematical system in which numbers vary continuously instead of shifting between discrete states as a digital system does. For example, in an analog clock, the hour, minute, and second hands all move continuously. Halfway between 2:00 and 3:00, the hour hand will be pointing halfway between the 2 and the 3. In a digital clock, the numbers simply shift between discrete states: halfway between 2:00 and 3:00, the hour digit will still read 2.
A slide rule is an analog calculator whereas an abacus is a digital calculator because the slide rule's numbers vary continuously whereas the abacus' numbers are either all the way up or all the way down, there is no meaning for a bead on an abacus that is half way up or down. Analog televisions are operated by machinery where the numbers vary continuously, whereas in digital television the numbers shift between discrete states; this is also true of analog computers and digital computers.
Disadvantages of Analog Systems
The need to make fine measurements when reading analog devices severely limits their practical accuracy, and digital devices are usually much more accurate in practice.
Advantages of Analog Systems
On the other hand, analog devices usually have better resolution than digital ones, because digital systems must round the data to the nearest discrete state. For example, suppose the analog signal has values 2.91, 2.99, and 3.04. If the digital system only has room to store integers, all three values will be stored as 3. A digital system with more possible states - for example, each tenth rather than simply each integer - can mitigate this problem, as 2.91 will round down to 2.9 instead of up to 3. However, it cannot completely solve this problem: 2.99 and 3.04 will both still round to 3.
Digital systems can also only measure data at discrete times. We do not know what the signal - for example, the sound wave - is doing between the measurements. More than one possible wave can fit. This problem is called aliasing; it can also be mitigated by taking more frequent measurements but never totally solved.