Earth's axis is tilted by approximately 23.5 degrees. In other words, Earth's daily rotation is shifted by 23.5 degrees with regard to its yearly revolution around the sun. This axial tilt is the reason why Earth experiences different seasons throughout the year, and also why summer and winter occur opposite each other on either side of the equator -- and with greater intensity farther away from the equator.
The sun burns with the same intensity all year. Earth's elliptical orbit brings it closer or farther at different times of year, but this change in distance has a negligible effect on weather. The important factor is the incident angle of sunlight. As an example, imagine that you have a flashlight and a piece of paper. Hold the paper so that it is perpendicular to the beam of the flashlight, and shine the light on the paper. The light hits the paper at 90 degrees. Now, tilt the paper. The same light is spread over a larger area, and is therefore much less intense. The same phenomenon occurs with Earth and the sun.
Equator Versus the Poles
The reason the equator is the hottest part of the planet is because its surface is perpendicular to the sun's rays. At higher latitudes, however, the same amount of solar radiation is spread over a larger area, due to Earth's spherical shape. Even without any tilt, this would result in the equator being warm and the poles being cold.
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Because Earth is tilted, different latitudes receive different sun angles throughout the year. During summertime in the Northern Hemisphere, Earth is tilted so that the Northern Hemisphere is angled more directly at the sun. It receives more direct sunlight and is warmer. At the same time, the Southern Hemisphere is angled away from the sun, so it receives less direct sunlight and experiences winter. The axial tilt doesn't change throughout the year, but as Earth travels to the other side of the sun, the opposite hemisphere is angled toward the sun and the seasons change.
Length of Days
At the fall and spring equinoxes, in mid-September and mid-March, the axis is pointed neither toward nor away from the sun, and the Northern Hemisphere and Southern Hemisphere receive the same amount of sunlight. Day and night are of equal length at these times. After the equinox, the days begin to get shorter in one hemisphere and longer in the other. At the summer and winter solstices on the 21st or 22nd of June and December, the days are at their longest or shortest, respectively. The summer solstice in the Northern Hemisphere, June 21st or 22nd, is also the winter solstice in the Southern Hemisphere, and vice versa.