Think of it this way:  nutrition is the food, nutrients are the leftovers after everything's eaten.  Simple as that.

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A real reef, and the basic idea of a reef tank, is to maximize nutrition for good fish and coral growth, but minimize nutrients in order to reduce nuisance algae growth. Skimmers, which are "protein" skimmers, only remove nutrition, which is great if all you have is fish; but skimmers have no effect on nutrients because nitrate etc does not stick to the air bubbles. Matter of fact if you took a fresh batch of newly made saltwater and put a skimmer in it, and then added pure nutrients, the skimmer would not have any skimmate at all even though nitrate, phosphate etc were sky high. However if you added nutrition (phyto, plankton, ground up flakes, coral food, etc) to that same batch of saltwater, the skimmer would go crazy.

Algae, however, do consume nutrients, and would start growing out of control in this high nutrient saltwater. That's why a tank can have lots of algae, but low nutrients: because the nutrients are all consumed by the algae. You have to remove this algae from the tank however.

Another way to understand this is to think about cooking a steak in your kitchen. The steak (nutrition) is what you want, but the smoke (nutrients) is what you don't want. You want to keep the steak in your kitchen (keep the nutrition in your water), but you want to remove the nutrients; so you use an exhaust fan to remove the smoke, and you use algae to remove the nutrients.

The word "nutrient" and "nutrition" are commonly confused when talking about reefs. "Nutrition" is just like food; it feeds the fish and the corals and the microbes. Nutrition include fish waste, coral slime, leftover food particles, algae (yum sushi), vitamins, and amino acids. This nutrition can be in the form of pieces or particles that you can see, or very very small particles that you cannot see call dissolved organic carbon ("DOC") . All this nutrition feeds some animal or microbe somewhere in your system, and it is the carbon in this nutrition that is being sought.

"Nutrients" however are different; they are Nitrate, Nitrite, Inorganic (Ortho) Phosphate, Ammonia, and Ammonium, i.e., the things your test kits test for. There is no carbon in these; just nitrogen, phosphorus, oxygen and hydrogen. Nutrients are what remain after the food chain does it's work, all the way down to the last bacteria. This is also called "re-mineralization", because all the organic particles have had the carbon removed such that only the "minerals" (nitrogen etc) remain.