When someone asks "What's your sign?" they're only asking about your Sun sign - just one piece of your astrological puzzle. Your complete birth chart contains the positions of the Moon, Mercury, Venus, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto.
Each planet represents different aspects of your personality. Your Moon sign reveals your emotional nature, Mercury governs communication, Venus shows how you love, and Mars indicates how you take action.
This is why people of the same Sun sign can be dramatically different. A Gemini with Moon in Cancer will be much more emotional than a Gemini with Moon in Aquarius.