The Horoscope

The horoscope is a chart showing the positions of the planets, Sun, and Moon in the sky at a person's birth. Their positions are located in the zodiac---a narrow belt of constellations centered on the ecliptic. The ecliptic is the path the Sun takes through the stars throughout the year (as opposed to the arc it travels from sunrise to sunset). The zodiac is divided into 12 signs named after the constellations through which the Sun, Moon, and planets passes. Your "sign" is the zodiac sign which the Sun was in at your birth.
Right away you run into a problem with the zodiac constellations---some are large (like Scorpio or Virgo) and others are small (like Aries and Cancer). Because the rate that the Sun moves along the ecliptic is nearly constant, the Sun spends more time in the large zodiac constellations than in the small ones. It does not matter whether you use the ancient constellation boundaries or the modern boundaries recognized by the International Astronomical Union (though, the IAU boundaries have the Sun spending part of its time in the non-zodiac constellation Ophiuchus!). However, the dates listed in the newspaper for the horoscope signs are all 30 or 31 days long (even for tiny Aries) and the horoscopes do not include the constellation Ophiuchus. (However, the astrologers can simply counter that with their constellation/zodiac boundaries are different from the astronomical ones.)
Because of an effect called precession, the zodiac constellations slide westward along the ecliptic, making a complete circuit in about 26,000 years. Since the zodiac signs were named over 2000 years ago, the stars have moved by about 1/12 of the zodiac (about one sign's worth). Your "sign" may be one month off! (The different sizes of the constellations prevents me from making a more definite statement.) For example, if your sign is a Sagittarius, then the Sun was actually located in the constellation Scorpio when you were born. Actually, for part of the Sagittarius timeframe, the Sun is in Ophiuchus, so perhaps that is not a good example. (Some astrologers today do take precession into account all the while continuing to use the old labels for the sun signs for the sake of their clients---the subjectivity, training, and cultural context of a given astrologer make it hard to make a blanket statement.)
The horoscope includes the position of each planets in the zodiac and where they are with respect to the person on the Earth at the time of his/her birth. Because of this, creating a horoscope is a bit complicated. There are some standard rules (most of which have not changed for thousands of years despite the dramatic improvements in our knowledge of how the planets and stars move), but how much emphasis an astrologer will give to each rule in developing the horoscope, depends on the creativity of the astrologer. This lack of objectivity is one reason why astrologers cannot agree on the right prediction for any given person. Unlike astronomy, astrology does not have clear objective observations of nature (experiments) determine the truth.
The recent (December 2010/January 2011) furor over the astrology signs being a month off is not the problem with astrology despite what the mass media has latched onto. Some modern astrologers have taken precession into account (check with your favorite one) when they cast their horoscopes for their clients and they do not use the constellation boundaries set by the IAU in the 1930s. So an astrology "constellation" or "zodiac sign" doesn't match the astronomy constellation. If you believe what has been said in the mass media, then it would be a simple matter of just re-labeling a person with a different sign and just continue using the horoscope to predict the future or personality of someone. The real problem remains.