Is my Moon sign more important than my Sun sign?

That depends on the tradition. Western astrology tends to lead with the Sun, Vedic astrology with the Moon.

Does the Moon phase at my birth matter?

Some astrologers read it. It is a separate factor from your Moon sign.

Why does my horoscope say "the Moon enters Leo"?

That is the moving Moon. It describes the sky that day, not your chart.

Can two people born on the same day have different Moon signs?

Yes, if the Moon changed sign that day.

What does my Moon sign mean, and why does the Moon change sign so often?

Your Moon sign is the zodiac sign the Moon was in at the moment you were born. It never changes in your chart. The Moon in the sky does keep moving, entering a new sign roughly every 2 to 2.5 days. In astrology, the Moon sign is read as your emotional side: needs, moods and habits.

Why does the Moon change sign so often?

Because it moves fast. The Moon circles the Earth in about 27.3 days, measured against the stars. NASA's StarChild site puts it this way: "The Moon appears to move completely around the celestial sphere once in about 27.3 days as observed from the Earth." (NASA StarChild)

The zodiac has 12 signs. Divide 27.3 days by 12 and you get about 2.3 days in each sign. The Moon's speed varies a little through its orbit, so the real stay is roughly 2 to 2.5 days.

Compare that with the Sun, which spends about a month in each sign, or Saturn, which stays in one sign for a couple of years. Apart from the rising sign and the houses, the Moon is the fastest-changing part of a birth chart.

What is the difference between the moving Moon and my natal Moon?

These are two different things that share a name, and they are easy to mix up.

Your natal Moon is fixed. It is where the Moon was at your birth, frozen in your chart for life. When someone asks "what is your Moon sign?", this is what they mean.
The moving Moon (the "transiting" Moon) is where the Moon is in the sky today. It changes sign every couple of days and is the same for everyone.

So "the Moon is in Scorpio today" does not change your Moon sign. If your natal Moon is in Gemini, it stays in Gemini. Some astrologers read the moving Moon against your natal chart day by day, but that is a separate practice from reading your own Moon sign.

What does a Moon sign mean in astrology?

In Western astrology, the Sun is read as your core identity and the Moon as your inner, emotional side: what comforts you, how you react, what you need to feel safe. A Moon in Taurus, for example, is traditionally linked with a need for steadiness and comfort. A Moon in Aries is traditionally linked with quick, direct reactions.

Vedic astrology leans heavily on the Moon. The Moon sign is called the rashi, and the Moon's position is the starting point for the timing systems called dashas.

These are traditional meanings, not tested personality facts. There is no good evidence that the Moon at your birth shapes your emotions. You can still use a Moon sign description as a prompt: does this fit me, and if not, what does? That kind of reflection can be useful even without believing the sky caused anything.

Can my Moon sign be wrong?

Yes, if you were born near the moment the Moon changed sign.

A worked example

Say the Moon moved from Gemini into Cancer at 9:00 pm on a Tuesday.

Jordan, born that Tuesday at 7:30 pm, has a Gemini Moon.
Riley, born the same Tuesday at 10:15 pm, has a Cancer Moon.

Same date, different Moon signs. A website that asks only for your birth date, with no time, has to guess. If you don't know your birth time and were born on a change day, your Moon sign is genuinely uncertain. Check the day in a calculator and see whether the Moon changed sign at all. If it didn't, your Moon sign is known even without a time.

Two more things can change the answer:

Zodiac. Your Vedic (sidereal) Moon is often one sign earlier than your Western (tropical) Moon, because the two zodiacs are about 24 degrees apart.
Time zone. Entering the wrong time zone, or forgetting daylight saving time, shifts the birth moment by an hour or more.

How do I find my Moon sign?

Find your birth date, place and, if you can, the exact time.
Enter them in a chart calculator such as Astro.com.
Check which zodiac it uses. Compare with other tools only when the zodiac is the same, and for Vedic, the same ayanamsa.
Note the Moon's degree, not just the sign. A Moon at 0 or 29 degrees sits close to a sign change, so the birth time matters more.

How does Mystogian work out your Moon?

Mystogian calculates your chart with astronomy code on our server, from your birth date, time and place. The AI never calculates planet positions. You can switch between the Western (tropical) and Vedic (sidereal) chart, so you can see your Moon in both. The app also accepts an unknown birth time.

FAQ

Is my Moon sign more important than my Sun sign?: That depends on the tradition. Western astrology tends to lead with the Sun, Vedic astrology with the Moon.
Does the Moon phase at my birth matter?: Some astrologers read it. It is a separate factor from your Moon sign.
Why does my horoscope say "the Moon enters Leo"?: That is the moving Moon. It describes the sky that day, not your chart.
Can two people born on the same day have different Moon signs?: Yes, if the Moon changed sign that day.

Sources

NASA StarChild, "What is the orbital period of the Moon?": https://starchild.gsfc.nasa.gov/docs/StarChild/questions/question32.html
Astro.com free chart calculator: https://www.astro.com