Quick answer
Your Moon sign is the zodiac sign the Moon occupied in your birth chart.
In modern beginner Western astrology, the Moon sign is commonly used for emotional style, inner needs, comfort, memory, instinct, and private reactions. It is one layer of the chart, not a complete description of a person.
The Moon moves much faster than the Sun, so a birthday alone may not always be enough. A birth chart calculator usually asks for birth date, time, and place to place the Moon more accurately.