There are quite a few ways to cook carrots, and the best way depends a great deal upon the method you prefer or what you'll be using them for. Two of the simplest methods for cooking carrots are steaming and boiling, both of which include water and both are quite fast. Carrots can also be cut up and roasted in an oven, which is a dry form of cooking that can caramelize some of their natural sugars. You can cook carrots by sautéing them in oil or butter, often using water to boil or steam them first.
The best way to cook carrots depends a great deal upon whether you prefer to use wet or dry heat in your cooking. Boiling is, perhaps, the easiest way to prepare carrots. You can make them by simply cutting them into even pieces, either as slices or small chunks, and then placing them in a pot of boiling water. The carrots cook in a short amount of time, but the boiling water can extract nutrients from them.
Steaming carrots is often preferable, since the steam does not take away the nutrients that boiling water does. A steamer basket can be used to cook carrots, by placing slices or chunks of carrot into the basket and putting it in a pot of boiling water. The water should not actually touch the carrots, and the heat from the steam cooks them. These are both wet cooking methods that are fast and simple.
An oven can be used to cook carrots through the application of dry heat commonly referred to as roasting. Simply cut up the carrots into even sized pieces, to ensure they all cook at the same time, and place them on a baking sheet in the oven. You can toss them with some oil and salt and pepper, though this is not necessary. Simply turn them every once in awhile as they roast; the dry heat can sweeten the carrots by bringing out their natural sugars.
You can also cook carrots through something of a combination of both dry and wet heating methods, such as sautéing. Place a small amount of water in the bottom of a pan and, some butter and the carrots, and cover the pan. As the butter melts, it will begin to cook the carrots along with the simmering water. Once the carrots are mostly cooked, then remove the lid from the pot and allow the water to boil off in steam. The buttery carrots cook slightly longer to caramelize, and you can add some brown sugar to taste.