Examples of continuous, quantitative variable: your weight over time. You could weigh 155 pounds today, 155.8 tomorrow, and so on. So: Your weight is a continuous variable, a quantitative variable because measurement is involved, and one of the proper units of measurement is pounds (English system).
Example of a qualitative variable: make of car. Your mom has a Ford and your dad a Mercedes-Benz. Do we measure "make of car" with numbers? No. Does your cousin have a car which is half Ford-built and half Mercedes-built? No, so "make of car" is discrete; it's this or it's that, but nothing in between makes sense. Unit of measurement? None.
Sex (male or female): Despite today's discussions about transgender individuals, it's long been accepted that individuals are either male or female, a discrete designation. Those are qualitative labels. No unit of measurement here.
Continuous vs discrete is if you can count vs. measure the results. For example: you can run 13. 5 miles but you can't have 13.5 dogs. Miles (measurable) are continuous while dogs (countable) are discrete.
Qualitative results are when a result is not a number, and qualitative is when the result is a number. For example: if you're doing a lab and a result is either going to be "blue" or "green", that's qualitative, since those aren't number values. However, if you were measuring distance, that would be qualitative, since you would get a result of "6 meters" or "2.5 inches", which are numerical values.
The scale of measurement are the units in which you are measuring something it. For example: distance has units of inches, feet, miles, etc... and weight has units of grams, kilograms, tons, etc...
