The ancient Greeks thought that the seasons owed their existence to an unfortunate marriage between Persephone, the daughter of the goddess of the harvest, and Hades, the god of the underworld.  For four months out of the year Persephone would travel to the underworld to visit her husband; stricken with grief, her mother Demeter would let the fields grow cold and lifeless.

The actual reasons may not be quite as dramatic, but certainly are just as interesting.

To start with, the Earth's orbit is an ellipse.

An ellipse is a slightly flattened circle.

The Earth's orbit around the Sun can be seen in the diagram to the left.  Notice the elliptical path the Earth takes in its journey around the Sun.

Because the orbit is an ellipse, that means that at some point, the Earth travels slightly closer to the Sun during its 365 day journey.  The Earth reaches the point in its orbit closest to the Sun in January, and it reaches the farthest point from the Sun some six months later, resulting in a slight change of distance from the Sun.  If that were all that governed weather, we'd have summer in January, and Winter in July! This may be true for our Southern Hemisphere friends, but not up in the North. Something else must be going on.

It ends up that the largest contributor to the change in seasons is the 23 degree tilt of the Earth on its axis, the imaginary line that runs through the middle of the earth from the North Pole to the South Pole.  The change in distance of the Earth to the Sun is a very minor player.

Because the direction of the Earth's tilt changes in relation to the Sun, the northern and southern halves of our planet get differing amounts of sunlight over the course of the year.  In the Northern Hemisphere of the Earth the axis points most toward the Sun in June.  Because the Earth is leaning toward the Sun, it receives direct rays of sunlight and is warmer, as seen in the image below.
 

On the other hand, the Earth's axis points most away from the Sun in December.  Because the Earth is leaning away the sun, it receives the least amount of direct rays of sunlight and is cooler, as seen in the image below.
 

But wait, there's more.  In the summer, the angle of the Sun is higher, and therefore the days are longer. This gives the Sun more time to heat the Earth, so it gets hotter.  In the winter, the angle of the Sun is lower, and the days are short, giving the Sun less time to heat the Earth. This is a secondary effect.

Make sense yet?  Try thinking about it this way.  When the Sun is higher in the sky, the light is falling straight on you, and so more light (and more heat) hit each square centimeter of the ground.   When the Sun is low, the light gets more spread out over the surface of the Earth, and less heat (per square centimeter) can be absorbed.  Since the Earth's axis is tilted, the Sun is higher when you are on the part of the Earth where the axis points towards the Sun, and lower on the part of the Earth where the axis points away from the Sun.

Because the Earth's orbit around the Sun does not change, we can predict the seasons;
 

The summer solstice is the first day of summer and the longest day of the year.  Summer solstice occurs in the Northern Hemisphere when the North Pole is leaning more directly toward the sun than it does on any other day of the year.  During the summer season, the Northern Hemisphere is warmed by more direct sunlight and days are long and hot.

The winter solstice, by contrast, is the first day of winter and the shortest day of the year.  As you might have guessed, the winter solstice in the Northern Hemisphere occurs when the North Pole is leaning away from the sun.  When the North Pole is pointing away from the sun, the Northern Hemisphere receives only indirect sunlight; that is why winter is so much colder than summer.  Brrrr!

 

Equinoxes, on the other hand, occur during transition periods when the Sun shines directly on the equator.  These days are marked by equal periods of light and darkness.  The autumnal equinox is the first day of autumn and occurs when the North Pole begins to lean away from the Sun; the vernal equinox is the first day of spring and occurs when the North Pole begins to lean toward the Sun again.

The distance of the Earth to the Sun is a smaller effect yet, but it does exist! So the Southern Hemisphere gets slightly hotter summers and slightly colder winters than the North.  But only by a couple of degrees, and only on average. Your mileage may vary!
 
 

And, if all of that wasn't enough, here is one more thing to think about!  The days are longer in summer and shorter in winter the further you move from the equator. It's generally dark on a summer night in Florida by 8:30 p.m., but in Vermont, there will still be some light at 10:00 p.m.  The situation is reversed in the winter, where the Sun will go down in Vermont by 3:45 while it remains light in Florida until 5:15.   Northern Alaska is called the "Land of the Midnight Sun" because it never gets dark during their summer months.  That part of the Earth is facing the Sun all day and all night.  Antarctica never sees daylight during those months.  Let's hope the penguins have nightlights!  Of course, the situation is reversed in December and January when northern Alaska never sees the Sun and Antarctica sees the "Midnight Sun".
 

The picture on the left was taken at 9 p.m. Eastern Standard Time on December 20.  Notice that it is dark in North America, South America, Africa and Europe, while it is daytime in most of Asia and Australia.  You can see that Alaska has been in darkness all day, while in Antarctica, the day never ends.

Credits
http://www.electronicpassport.com/601-seasons.html
http://www.nationalgeographic.com/xpeditions/activities/07/season.html
http://www.nationalgeographic.com/xpeditions/lessons/07/g35/seasons.html
http://www.badastronomy.com/bad/misc/seasons.html