A quantitative variable is measured numerically. With measurements of quantitative variables you can do things like add and subtract, and multiply and divide, and get a meaningful result. In the previous example, "Age" was a quantitative variable.

These allow for classification based on some characteristic. With measurements of qualitative/categorical variables you cannot do things like add and subtract, and multiply and divide, and get a meaningful result. In the previous example, "Gender" was a qualitative/categorical variable. Gender was categorized as either male or female.

A discrete variable is a quantitative variable with a finite number of values. For example, imagine you rolled a six-sided die four times and measured how many times you rolled an even number. What are your possible outcomes? {0, 1, 2, 3, 4}

A continuous variable is a quantitative variable with an infinite number of values. Take temperature for example. Temperature can take on an infinite number of values, such as 80 degrees, or 80.01 degrees, or 80.0050592359 degrees. In the previous example we were limited to a finite number of values (you couldn’t roll 1.5 even numbers), which is what made it discrete.