The Sun does not belong to any constellation. This is because our Earth goes around the Sun. As a result, the Sun moves in the sky relative to the other stars. So, the Sun appears to move through the constellations of the zodiac, which is why you hear that the Sun is in a particular zodiac constellation in a particular month. For example, in September, the Sun is in the constellation of Virgo. In October, it will go to the constellation of Libra, and so on.
You can easily understand this if you draw the picture of the Sun with the Earth orbiting the Sun. Now all the other stars are essentially at infinite distance (the nearest other star is at 4.3 light years which is much farther away from the 8 light minutes at which the Sun is located). Now, you can see the relative position of the Sun with respect to the stars and see how it changes as the Earth goes around the Sun.
Added by Karen: Also note that the constellations do not represent physical volumes in our galaxy, but rather are used to divide the sky as seen from Earth into different areas. Stars which are in the same constellation are usually not physically associated at all, some of them are much closer than others, but just happen to lie along almost the same line of sight.
This page was last updated on July 21, 2015.