Answer:
Yes, Aircraft can fly below, in, or above the clouds. There are special qualifications and rules that apply. Some clouds are low, some are higher. The aircraft must be capable of climbing to an altitude that will get them above the clouds.
Most small single engine airplanes have a 'ceiling' of less than 15,00 feet, larger singles about 25,000 feet and jets around 40,000 feet. So the ability depends on the height of the clouds and the capability of the airplane.
Flying in, through, or even very close to the clouds requires special training and aircraft capabilities. This is referred to in aviation as "Instrument Flight Rules" (IFR), as opposed to the other type of flight where only visual reference outside the airplane is used "Visual Flight Rules" (VFR).