The four building blocks
Every chart is made of four things. Planets are the actors: each one stands for a kind of drive, like feeling, thinking, loving, or acting. Signs describe the style each planet works in. Houses describe the area of life where it shows up. Aspects are the angles between planets, and describe how they get along.
So “Venus in Leo in the fifth house, square Saturn” reads as: your way of loving (Venus) is warm and expressive (Leo), shows up most in romance and play (the fifth house), and has to negotiate with caution or responsibility (the square to Saturn).
The Big Three
Most people start with three placements. The Sun sign is the one in horoscope columns, and describes how you grow into yourself. The Moon sign describes your emotional life and what helps you recover. The Rising sign, or Ascendant, describes your manner and how you begin things. The Sun depends on your birth date, the Moon on the date and sometimes the time, and the Rising sign on the exact time and place. You can look up the last two with the Moon sign and Rising sign calculators.
The planets
Mercury describes how you think and talk. Venus describes affection, taste, and what you value. Mars describes drive, desire, and anger. Jupiter and Saturn are the social planets, describing growth and structure. Uranus, Neptune, and Pluto move so slowly that whole generations share their signs, so they matter most where they touch your personal planets or fall in a house.
Houses
The twelve houses divide the sky into areas of life, from self and money through home, love, and work to the hidden twelfth house. Houses depend on your birth time. Without one, a chart can still show every planet's sign, but not the houses. Read more in the twelve houses.
Aspects
Aspects are the conversations between planets. Trines and sextiles flow easily. Squares and oppositions create friction, which is often where people grow. Conjunctions fuse two planets into one voice. See planetary aspects for the full list.
What a chart can and can't tell you
A birth chart is a symbolic language, not a scientific prediction. It can give you words for patterns you recognize and questions worth asking. It can't tell you what will happen, who you should marry, or what you're capable of. The most useful way to read a chart is to test it against your own life: keep what fits, and hold the rest lightly.