Counter is when you draw a line horizontally through your hips and a line at right angles to your hips points outside the direction of the turn. This is also called counter rotation.
In the camps they will set you up with a hip-o-meter. You make a hip-o-meter by taking two ski poles and reversing one to the other, and one goes on your front waist and the other goes on your lower back and the ends are hooked into the wrist straps. This gives you your horizontal line as a reference point.
In early counter at the top of the turn, you point your hips to the outside of the turn. This is completely opposite of facing your hips down the hill into the next turn which will set your hips up opposite of any counter rotation.
One of Jay's posts on this is in the classic thread's section. But this is not the one with the link to the movie. I'll look for that later and pop a link up - but time to get off the forum and earn some money.
Strong tipping of the inside foot will help to set yp this counter rotation and hip drop. This increases your edge set and leaves you very skeletally strong.
If your turn speed is very low, this counter rotation is mixed with counter balance. Counter balance is when your body is in a c-shape. In the early part of the turn if your uphill shoulder is tipped to match the slope then you will not have any counter balance. If your shoulder is kept down at that part of the turn you'll have more counter balance.
Search Jay's posts. He has an excellent post of both of these with about a 15 minute video of HH explaining and showing these. Search Skiersynergy (jay)'s posts or look for the word counter and bring up Jay's posts.