Jen Brown's excellent answer just about covers most of it. I urge all parents to pay especial note to her answer. Barbara Carleton's answer covers much of the rest.

I jointly run a private school with my wife. Here is the most commonly observed mistake from an educator's point of view.

Allowing your child to give up easily

Learning is a messy, organic process. Everyone learns at a different pace. One common factor in learning anything however, is that hard work is unavoidable. Here's a graph every parent should know:

Forecasting Confidence Levels With the Bipolar Learning Graph

This graph applies even to adults.

Whenever you begin learning a new subject such as maths, science, programming, a new language, or the piano, your enthusiasm level soars. Basic concepts are simple and easy to grasp, everything is fresh and new = this is fun!

Then comes the trough where simple things, just aren't so simple anymore. Concepts become more complex and harder to understand, everything is old hat = this is not fun anymore.

With work and perseverance, you eventually master the more advanced material and pass the inflection point, and gain confidence. Advanced concepts become easy, you get to apply the knowledge in new and interesting ways of your own choosing = this is way more fun than it was before!

When the going gets tough before the inflection point, many children tend to push back. They scream. They cry. They refuse to go to school. They throw tantrums. At this point, and out of misplaced protectiveness, many parents simply, well, give up.

I have never met a so-called child prodigy who did not experience this trough or did not have a lot of hard work and perseverance behind his/her achievements. Not one. The idea that natural talent is all you need to achieve your goals and that natural talent means everything will come easy is false, but many parents still hold on to it.

The trough is a natural and inescapable part of human learning. What's more, it's not just that one trough, but a series of troughs in any given discipline.

Our job as parents is to guide and shape our children's development, boost them up during the high points, and support them as they work through the low points.