I'm surprised to see that there is such a debate concerning types of fats. I don't think there are really different opinion camps on this. The following is not nuance. It is just the same information that you'd get if you asked any personal trainer or bodybuilder this question.
Saturated fats are from pigs, cows, and egg yolks. Saturated fats are saturated with hydrogen atoms. They are a very stable molecules that do not react very readily in the body, and are more difficult to access for energy, so they are more likely to be stored for later use. These can be stored where ever your body stores fat and can also be stored along artery walls, and can cause heart disease. These types of fats should be limited.
Fats in red meat and pork products should be limited, but that does not mean it's necessary to eliminate all red meat and pork products rather limit the fat content by choosing lean cuts of these meats. Also remember that dairy products that are not fat free contain a considerable amount of saturated fats. Primarily cheese, even white cheeses contain a lot of saturated fat and should be minimized. Although some people eat cheese as a protein source, I do not believe it should be considered as such. The amount of calories from fat in cheese is well over half even in a low fat cheese.
Egg yolks have notable amounts of cholesterol in addition to saturated fat. Cholesterol has been credited with increasing levels of testosterone, so it may be worth taking in a moderate amount for this purpose.
Unsaturated fats are from fish and plant sources. They are called essential fatty acids because your body does not make them on its own, so you have to take them in through your diet. Unsaturated fats have fewer hydrogem molecules than saturated fats, so they will contain one or more double bonds which makes them a polarized molecule thus more reactive in the body. This means your body can break it down more readily to use it for energy. Polyunsaturated fats are unsaturated fats that have 2 or more double bonds. In addition, unsaturated fats can combine with saturated fats and pull away a hydrogen atom converting it to an unsaturated fat. In this manner unsaturated fats can help you both burn stored fat and break down fatty acids in the blood stream for a healthier cardiovascular system.
I suggest taking in unsaturated fats at least once or twice a day. The best sources are nuts and plant oils including peanuts, almonds, cashews, sunflower seeds, olive oil, sunflower oil, safflower oil, flax oil, etc.
You'd be hard pressed to eliminate saturated fats completely from your diet because most unsaturated fats have some amount of saturated fats in addition and vise versa. You just have to read labels and consider how much you are getting of one compared to the other.
I should also include a note about trans fats which are a chemically created form of saturated fats. Trans fats should not just be limited but avoided completely. They have been proven to not only increase bad cholesterol, but to decrease good cholesterol. Trans fats can be found in icings, cooking lard like crisco, and many processed foods because they also act as a preservative.