Food contains both nutrients and calories (energy) among other things such as fiber and water. Calories come from carbohydrates, fats, and proteins. Nutrients are things such as vitamins, minerals, and phytochemicals (chemical compounds produced by plants).
Nutrient Density is the amount of nutrients you get from food given the amount of calories (energy) that food contains.
As a simple mathematical formula:
Nutrient Density = Nutrients / Calories
A more nutrient dense food gives you more nutrients with fewer calories.
Dr. Fuhrman's Aggregate Nutrient Density Index (ANDI Scores) shows how popular foods score in terms of nutrient density per calorie. As expected, Kale and dark leafy greens are most nutrient dense.
It should be noted that Nutrient Density looks at food in a particular way using a strict formula that compares nutrients to calories and should not be regared as an indicator of that foods overall quality or benefit in your diet. It is simply a number representing how nutrient rich a food is in relation to how much energy it gives you.