Continuous data do not fall into discrete categories. Instead they produce a continuous distribution. Some of these distributions can be modeled algebraically. The two most common ones are Poisson and Gaussian (or normal).Examples of continuous traits include yield, size, nutrients, etc. Certainly you could score these traits on a scale, but if you measure the exact quantities, they will not fall into clear classes.
A perfect normal distribution (top) as compared to a histogram of yield data from a QTL experiment (bottom). Data points fall into a continuous range, not discrete classes.
Top image from http://mathworld.wolfram.com/PoissonDistribution.html, lower image from T. Fulton using QGene software (http://www.qgene.org).