This is such a noob mistake, and likely experienced cooks will scoff at this article, but it needs to be said that the pot lid makes a big difference. So much so that I got myself pot lids for my pans that didn't have them.
If it isn't obvious, the pot lid keeps heat in, and it matters. At the end, we will also discuss examples when you definitely do not want to put a pot lid, but here are cases when you want it on.
Boiling Water
This was pretty obvious to me, but it merits to be said, boil water with a lid on. It matters first because it lessens the escape of water. Assuming you actually need the liquid you're boiling, like for a soup or as the medium where you want to cook pasta, the pot lid acts to contain the water. Steam condenses on the under side of the lid and drips back in. Second, it lessens the time to boil the water. This is a big consideration for me. Basically, it's the water vapor that has the most energy (exactly why it can escape the surface tension of the liquid water), so keeping as much in the pot by covering it keeps the heat (our energy) in. This is the same for any air between the lid and the surface of the water. Basically the heated air will rise along with its heat if you are boiling water unheated. Pot lids, by just having a small hole, reduces all of this. In my experience I cut boiling time by at least 25% with the lid on.
Sunny-side up eggs
Now, this was a surprise for me. Usually, if you want your yokes to be less runny, you cook your sunny side up eggs in oil of some depth, conceivably because you want the heated oil to cover the top part of the eggs. Use a lid on the top of your frying pan, keep the heat on low to medium heat, and your yokes will be less runny. We're using the same principle here, but it's great that it works with shallow pans. You keep any heated air between the lid and the pan and thus js precisely what you need to cook the top part where the yokes are. And it's the same case that you will end up using less electricity (the reason why you can get away with low to medium heat.
Vegetables and other greens
This is a bit like the boiling water case, but this is a special mention because of how at times, you want to cook vegetables thoroughly to make them less tough (think carrots or peas), but at the same time you don't want to dry them out. Steaming is the best method in these cases, but the alternative is to stick these in a pan with water and a lid. You get a similar steam effect, you'll notice the fogged up glass. And basically you avoid drying your greens out because you keep the moisture in, and you can cook them quicker because you keep heat in.
In summary...
In general, this works in a lot of cases where you want to speed up cooking or keep moisture in. When defrosting meats for example, Basta sa Bahay will always suggest that room temperature airing is the best (you use less electricity), in a pinch, low heat and a lid is one of the best solutions. You want to stay away from high heat as it runs the risk of making meat tough when overcooked, and the low heat under a lid speeds up the process. There are a ton of uses, so just make sure you have a lid for all the sizes of pots and pans you have.
Finally, just note cases where you definitely don't want a lid: cooking where you want to rid your dish of moisture. This means any crispy dishes (part of the crispy texture is the lack of moisture) and sauces that you want to reduce (in this case you mean for the moisture to rise up and away).