The Earth moves around the Sun in a roughly circular orbit. The plane that includes Earth's orbit is called the ecliptic plane. While the earth is orbiting around the sun, it is also spinning around itself. When it is spinning, exactly two points on the surface, known as the north and south poles, do not move. The imaginary line going through the two poles is called the axis of rotation.
Seasons are caused by the fact that the axis of rotation is not perpendicular to the ecliptic plane. If they were perpendicular, the Earth's spin and orbit would go in the exact same directions. Instead, they are off by 23.5 degrees. The axis of rotation is tilted towards the sun during summer and away from the sun during winter. Why is the Earth apparently tilted in different directions at different times of year? It is in fact tilted in the same direction all year, it's just that the direction towards the sun changes throughout the year.
[Image from Timezone.com]
The day when the axis of rotation is tilted furthest away from the sun is called the winter solstice. This year, the winter solstice is December 22. The day when the axis is tilted closest towards the sun is called the summer solstice, which is somewhere around June 21 or 22.
[Image from Penn State University - I'm not sure what's up with the smiley faces.]
The reason axial tilt causes seasons is twofold.
First, the power from the sun is not evenly distributed on the Earth's surface. If you look at the above picture, there is about an equal amount of power between each of the yellow lines. However, towards the top and bottom of the Earth, that same amount of power is spread over a larger area of Earth. Therefore, the power per unit area is smaller closer to the poles, and larger at the equator. The power peaks at the Tropic of Capricorn during the winter solstice and at the Tropic of Cancer during the summer solstice.
Second, the length of the day grows shorter in winter, and longer in summer. In the above picture, you might notice that most of the tropic of cancer is in the dark. This indicates a shorter day. Also note that the entire arctic circle is completely in the dark. This indicates that it is perpetually nighttime. During the summer solstice, the arctic circle is in perpetual daylight. If you stand at the north pole, there is exactly one night and one day every year.
Both the length of the day and amount of power from the sun affect the temperature and climate. Around December, the northern hemisphere has shorter days and less power from the sun. However, the southern hemisphere has longer days, and more power from the sun. The southern hemisphere has summer at the same time that we in the northern hemisphere have winter. And vice versa. The southern hemisphere, I imagine, doesn't get many white Christmases.
Now, some people mistakenly think that the reason for the season is the distance from the sun. The Earth's orbit is not exactly circular, meaning that there are times of year when the Earth is closer to the sun, and times when it is further from the sun. This does have a very small effect on the climate, but it is not nearly as large as the effect of axial tilt. In fact, the day when the Earth is furthest from the sun (called the aphelion) is around July 7--in the middle of the northern hemisphere's summer. The Earth is closest to the sun (this point is called the perihelion) around January 3.
So there you have it--third grade science, as explained by me. Next time, I'll talk about more advanced stuff like precession or something.