What is the difference between discrete and continuous traits?

A discrete trait is one which can only take a finite number of values (usually integer), such as how many members in your family. A non-integer set might be the answer to your age, in years, calculated to the nearest month.

A continuous trait is one which, IN PRINCIPLE, can take any value. In practice, the measuring instrument and recording will often result in the continuous trait being converted to a discrete one.