Microwave ovens by their nature heat unevenly. The actual microwave radiation produces wave patterns with more and less amplitude within the cooking chamber. (From a physics standpoint, these are "standing waves" produced by internal wave reflections.) This is sort of the equivalent of "hot spots" and "cold spots" in a pan.
Most modern microwaves contain some sort of rotating plate or other moving part to mitigate this unevenness by keeping the food in motion (and thus aligning different parts of the food with the "hot spots"). Over time, some parts of the food will also become very hot, and the heat from those portions will gradually distribute to the cooler portions of the food around them.
The reason I bring all of this up is that the primary advantage to cooking at a "lower power setting" in a microwave (whether it truly can change wattage or whether it merely cycles between 100% power and off, as dpolitt's answer notes) is to allow time for those hotter portions of food to distribute their heat to the cooler portions during cooking.
If you're heating something that has easy heat circulation (e.g., boiling water or other thin liquids), it doesn't much matter what setting you use, so use higher power for time efficiency.
But in many foods using a lower setting can:
- Allow time for the food to heat more gradually and more evenly
- Prevent parts of the food from getting overcooked (which could lead to sections being dried out or otherwise changed to an overcooked texture)
- For longer cooking times, it may also allow the "cold spots" more time to "catch up" and spend more time at a cooking temperature, which may be desirable for texture (e.g., if the food has to soften) or other reasons
- Sometimes prevent undesirable "behavior" of food in the microwave when overheated: e.g., exploding or "popping" of some parts of the food, boiling over of some liquids, etc.
In general, I'd say that there's rarely much harm in using a lower cooking setting in a microwave, other than potentially wasting some energy. As long as you lengthen the cooking time to compensate and cook the food completely, a longer time is fine. So, that's what I'd recommend if you're at all concerned about the power setting.
The only time to avoid a lower power setting is generally in foods where longer cooking times would also be avoided in other cooking methods. For example, when steaming vegetables, a longer cooking time will often result in mushiness, which may be undesirable -- whether done in a pot on the stove or in a microwave on low power. (Some cooking requires a minimum temperature to achieve normal results too, like frying. In that case, you might need to use a higher power setting. I don't tend to use my microwave for such things, so I don't have a lot of recommendations for that.)
All of that said, the primary reason to use a microwave over other cooking methods is generally speed. That's why most people tend to just use full power on microwaves for the vast majority of tasks anyway. If you're heating a food where the things listed above are unlikely to pose a major problem, using higher power may be fine.
Lastly, in many foods interrupting cooking by periodic stirring is actually more effective to distribute heat evenly than a longer microwave time at lower power. Thus, I mostly tend to use a lower power setting myself only in foods that can't be stirred or where I don't want to be bothered to take the food out and stir periodically over a longer cooking time.
A glass of water in the microwave will result in less microwave energy reaching the popcorn. I use a large glass of water (8 - 12 oz.) when I'm softening butter because I want the water to absorb most of the energy. For popcorn, you may get by with 2-4 ounces of water.
I've also found that the amount of oil makes a huge difference. I sometimes pop popcorn in a paper lunch bag. If I don't use any oil, I get the combination of burnt and unpopped you described. With a teaspoon of oil added to 1/3 cup of popcorn, results are a lot better. Perhaps you just need a little more oil to get great popcorn.
Best Answer
It may already be an implicit assumption, but I think that time to cook is probably not as important as how to cook. While the power setting does affect the time to cook, the importance of time is to control how hot the parts of the food get.
As with all cooking, there is a need to control temperature. Running the microwave at less than full power allows time for the heat to be conducted from the hot parts to the cool parts. It is similar to searing vs. sauteeing.
I think that the accuracy of most microwave oven's power settings are good enough (my own testing shows about 10%). I would be more concerned about how I wanted the item cooked. Some examples: