Discrete is a type of random variable which may take on only a limited set of integer values, such as 1,2,3,...,10. The list may be finite, or there may be an infinite number of values. A discrete random variable is to be contrasted with a continuous random variable. (Variables in Nominal and Ordinal Scales)

Continuous is a type of random variable which may take any real value over an interval. This is contrasted with a discrete random variable, which may take only a limited set of values. (Variables on Interval and Ratio scales)

Climate Variables (temperature, precipitation, wind, evaporation, etc.) fit the definition of random variables as each observation is different from the next, and there is a function that describes quantity of the variable for each observation.

For example, temperature has a continuous character (thermometer can take measurements anytime), whose amount at a given time depends (is a function of) on many factors. Temperature measurement could be classified as an example of a continuous random variable that is measured on interval scale.

Character of precipitation is another example of a random variable - either as a discrete variable: precipitation status (nominal scale: days with precipitation, trace of precipitation, no precipitation) at a given time or as continuous random variable bounded at zero when treated as amount of precipitation (variable at a ratio scale).

REFERENCES:

Ross Sh., 1994. A First Course in Probability, 4 ed., Macmillan College Publishing Company, Inc., pp. 126-176