What are Continuous Random Variables?
Continuous random variables are random variables can take values from an uncountable set, as opposed to discrete variables which must take values from a countable set. Another way to put this is that a continuous random variable must be sampled from a distribution that yields an everywhere continuous cumulative distribution function. As a result, the probability that a random variable yields a specific number (in its range) is zero, and thus typically intervals of values are considered to obtain meaningful probability scores.
How are Continuous Random Variables Used?
Continuous random variables are used to model stochastic processes that take values from a continuous range. Some examples might be modeling temperature in some region or stock prices. Although the latter (strictly speaking) does a have a countable range (prices are incremented by pennies), it is typically more mathematically convenient to use a continuous random variable.
In data science, a regression model tries to predict values for a continuous random variable (process) as opposed to a classification model which tries to predict values for a discreet random variable (process).