What do you mean by 'how long the calories last'? A Calorie is a Calorie. Doesn't matter where you get it from, it's the exact same amount of energy. Fats do have a higher energy density than other macronutrients (9kcal/g vs 4kcal/g for carbs and protein), meaning that you get more calories from 1g of fat than 1g of carbohydrates. So 10g of fats will, obviously, give you more energy than 10g of carbohydrates.
If you're talking about satiety, then there is ample evidence that protein is much more effective than fats or carbohydrates in that regard: http://www.ncbi.nlm.nih.gov/pubmed/18469287.
The person you're replying to could have meant that different foods have different glycemic indexes [1]. Essentially, while foods may contain the same calories as measured by a bomb calorimeter or similar, the rate at which those calories are broken down by your body into simple sugars for absorption by your blood varies. A "slow" calorie in that sense would be something with a low glycemic index, like nuts, which would cause a much more gentle rise in blood sugar.
There's evidence that a low GI diet can reduce the incidence of type 2 diabetes and heart disease.
This is technically true, but useless. A calorie, as you surely know, is a measure of increasing the temperature of water. Imagine cooking one pot on high, increasing the temperature as fast as possible, and another gradually over the course of hours, applying the same total increase in heat energy to both. Do you really think the contents would cook identically in both pots?
As leoedin eloquently described, glycemic index was what I meant; do the calories hit you gradually, or all at once? But biology is freakin' complex, so at some level these concepts are all just leaky abstractions.
Okay, from the GI point of view I see what you're saying. I thought you were referring to the satiating effect of fats vs other macronutrients from the phasing of you're initial response; please forgive my knee jerk reaction.
It depends on your dietary goals and expected physical activity, bur personally, I'd reach for for some quinoa or make a lentil-based dish as a pre-run meal instead of nuts.
Complex carbohydrates and fiber rich foods won't spike your blood sugar (and subsequently your insulin levels), so you can last for longer without eating more food.
If you're talking about satiety, then there is ample evidence that protein is much more effective than fats or carbohydrates in that regard: http://www.ncbi.nlm.nih.gov/pubmed/18469287.