The sky's a busy place, moving around the earth in complex and dynamic ways. Just kidding! That's what people thought before Copernicus came along and figured out that we are not the center of the universe (what a selfless Pisces!).
Each of the planets represents a different aspect of your personality and life.
Nevertheless, we still say that the "planets" (including the sun and the moon) are "in" certain signs, meaning that they appear, from where we're standing on the earth, to be in a certain part of the sky. (So, if you were born at the same time on Mars, you'd be a totally different sign.) Each of the planets represents a different aspect of your personality and life, and each of the signs has characteristics that will influence them.
Your "natal chart" is essentially a map of the sky over your head at your birth, and it's what astrologers use to map out all those placements. These charts are infinitely complex, but you can get some general ideas by looking at what signs your planets are in.
Don't freak out if you find a sign unflattering. Just play to your strengths (and work on your flaws)!
You probably know your sun sign; to find out the placement of your other planets, as well as your rising sign, input your birth day, time, and location at an astrology website, like astro.com, that calculates free natal charts. Note that you'll need to know your time of birth to get your rising sign. Don't freak out if you find a sign unflattering: all personality traits have good and bad sides. Just play to your strengths (and work on your flaws)!
Aries: Energetic, self-confident, courageous, playful, naive, contentious, impulsive, arrogant
Taurus: Graceful, stable, possessing good taste, materialistic, stubborn, determined
Gemini: Intelligent, witty, well spoken, nonconformist, shallow, undisciplined, jack of all trades and master of none
Cancer: Sensitive, loving, nurturing, emotional, tenacious, shy, touchy, moody, combative
Leo: Ambitious, magnanimous, loyal, romantic, optimistic, affectionate, domineering, inconsiderate, egotistical
Virgo: Practical, detail-oriented, efficient, thorough, cautious, critical, judgmental, fussy
Libra: Hard-working, social, romantic, collaborative, gracious, materialistic, indecisive
Scorpio: Intense, willful, emotional, sensual, dedicated, secretive, vindictive, suspicious
Sagittarius: Independent, idealistic, spiritual, optimistic, argumentative, blunt, narrow-minded, fanatical
Capricorn: Dedicated, hard-working, frugal, methodical, natural leaders, serious, condescending, cold
Aquarius: Loyal, honest, humanitarian, helpful, friendly, eccentric, independent, determined, dogmatic, impersonal
Pisces: Sensitive, empathetic, artistic, intuitive, spiritual, indecisive, lazy, careless, dependent
The Classical Planets
Sun: Ego, identity, core ideologies and values, creative drive, life's purpose
Moon: Emotional nature, personal needs and desires, memory, unconscious thought, maternal issues, habits, sentimentality
Mercury: Communication, expression, coordination, thought and reason, idea generation
Venus: Emotional values, aesthetic tastes, love in relationships, attraction, financial habits
Mars: Energy, aggression, survival instinct, impulsivity, sexual desire
Jupiter: Morality, optimism, luck, growth/expansiveness, sense of humor, generosity, indulgence
Saturn: Self-discipline, structure, time management, responsibility, commitment, self-awareness