Discrete scale definition:
Discrete data, like counts, are numeric data that have a finite number of possible values and can only be whole numbers. Discrete data arise from observations that can only take certain numerical values. Fractions are meaningless. In some situations, mathematical functions or calculations are not possible either.
Discrete variables are measured across a set of fixed values, such as age in years (not microseconds). These are commonly used on arbitrary scales, such as scoring your level of happiness, although such scales can also be continuous.
Discrete data can be used as ordered categorical data in statistical analysis, but some information is lost in doing so.
Discrete scale examples:
The number of children someone has: 1, 2, 3, etc. are possible, but 1.5 children is not meaningful.
Credit card number: The number is a discrete value, but cannot be used for addition or subtraction, etc.
Another classic is the spin or electric charge of a single electron. Quantum Mechanics, the field of physics which deals with the very small, is much concerned with discrete values.
Another example might be how many students were absent on a given day. Counts are usually considered exact and integer. Consider, however, if three absences make a suspension, then aren't two absences equal to 0.67 suspension?