Very simply, a cyclone is a kind of storm, and a storm is air which is moving quickly from one place to another. A storm begins when a mass of warm, moist air from the equator meets a mass of cold, dry air from the northern hemisphere. The two masses of air will not mix instead they form a front, which is just the name for the boundary where the two meet. The air in them continues to move, and the warm air rises above the colder air, becoming cooler as it does so. Then the moisture which was in the mass of warm air condenses and forms clouds.
Meanwhile, at the center of the storm, the air pressure begins to fall, and winds blow round this area of low pressure. In the northern hemisphere, the winds blow in the anti clockwise direction, and this means that the warm air moves north round the eastern side of the storm, and the cold air moves south around the western side. So a cyclone is just another word for a low pressure area. Such cyclones can be enormous, sometimes about a thousands miles in diameter.