The reason why seasons occur relates to the axial tilt of the Earth - that is, the degree to which the Earth is tilted at an angle as it orbits around the Sun. Because the Earth is tilted in space, during any given part of the year, some parts of the Earth receive more sunlight - and, thus, heat - than other parts. The most obvious consequences to the human observer are changes in the length of the day, as well as the progression of the seasons: summer, autumn, winter, and spring.
Earth's axis is tilted relative to its orbital plane by an angle of 23.5 degrees. Put in plain language, this means that the Earth is tilted over at an angle relative to the path through which it travels through space. This means that instead of the north and south poles being angled at 90 degrees relative to the path the Earth follows through space, instead the entire planet is effectively leaning over at an angle. Axial tilts are common among planets, but vary considerably. For example, while Jupiter has almost no discernible tilt, Uranus's tilt is so extreme that the planet essentially orbits on its side, with an axial tilt equal to about 177 degrees.
The tilt does technically mean that, for example, the northern hemisphere is very slightly farther from the Sun during its winter period than during its summer period. However, despite what many believe, this is not actually responsible for the changes in the seasons. Instead, the reason why seasons occur is that, because of Earth's axial tilt, the hemispheres receive different amounts of sunlight, and therefore different amounts of solar energy and heat, at different times of the year. The dramatic effects on Earth's climate result in the changes that we know as seasons.
To understand the process involved, tilt a small model of the Earth - that is, a globe - on its side. Recall that the Earth spins around its axis, a line running between the north and south poles; one full rotation is referred to as a day. However, this spin does not occur at a 90 degree angle relative to Earth's orbit.
For this reason, every year the north pole will spend some time angled away from the Sun, as well as some time angled toward the Sun. (The same is true of the South Pole.) When the northern hemisphere receives more sunlight, it grows hotter - resulting in a period of time we know as summer. Several months later, the Earth has moved so that the northern hemisphere is pointed away from the Sun. This means it receives less sunlight, and grows colder - resulting in winter. The same process occurs in the southern hemisphere, but at the opposite time; that is, when the northern hemisphere is experiencing winter because it is pointed away from the Sun, the southern hemisphere is experiencing summer because it is pointed toward the Sun.
The closer one gets to either pole, the greater the effect of the tilt on exposure to sunlight and therefore the more pronounced the effects are in terms of seasonal changes. The high Arctic and Antarctic experience considerable periods each year where they are pointed entirely away from the Sun and receive no sunlight, as well as periods where they are pointed toward the Sun and experience 24-hour daylight. Close to the equator, the axial tilt does not result in significant changes in exposure to sunlight, so that the equator does not experience significant seasonal changes in climate. Instead, equatorial regions tend to experience "wet" and "dry" seasons, because the changes in precipitation change more than the changes in temperature.