You can create an astrological chart for any moment in time. It's like stopping the cosmic clock, and studying the positions of the planets at that moment. Your birth chart is the moment you arrived into this lifetime. It tells you all about the energetic influences at play, and what life lesson you can expect.
Horoscope means hour watcher in Greek, and is another name for the astrological chart. But that term is used most often now to mean predictions based on astrology, like the popular Sun Sign horoscopes.
This is what an astrological chart looks like, though they vary in appearance. This is for the Dalai Lama of Tibet.