Why do Muslims start Ramadan on different days?

It depends on how each community confirms the new moon. Some wait for a crescent to be sighted locally, some follow a sighting from another country, and some use astronomical calculation. Scholars have accepted more than one approach for centuries, so a one-day difference is normal and no one's fast is wrong for following their own community.

The moon decides the month

Islamic months follow the moon, not the solar calendar. A month lasts 29 or 30 days, and the Prophet ﷺ said, "We are an illiterate nation; we neither write, nor know accounts. The month is like this and this, i.e. sometimes of 29 days and sometimes of thirty days" (Sahih al-Bukhari 1913). The Quran also says the moon's phases help people keep track of time:

يَسْـَٔلُونَكَ عَنِ ٱلْأَهِلَّةِ ۖ قُلْ هِىَ مَوَٰقِيتُ لِلنَّاسِ وَٱلْحَجِّ "They ask you [O Prophet] about the phases of the moon. Say, 'They are a means for people to determine time and pilgrimage.'" Quran 2:189

The Prophet's instruction

The Prophet ﷺ said, "Start fasting on seeing the crescent (of Ramadan), and give up fasting on seeing the crescent (of Shawwal), and if the sky is overcast (and you cannot see it), complete thirty days of Sha'ban" (Sahih al-Bukhari 1909). The crescent cannot always be seen: clouds, weather and the moon's position can hide it in one place while it is visible in another. That is the main reason for different start dates. See When does Ramadan start in 2027?

The three common methods

Local sighting: the community waits to see the crescent in its own region.
Following another country: many communities follow an announcement from a Muslim-majority country, for example Saudi Arabia, so that they start together with the wider Muslim world.
Astronomical calculation: some councils and mosques in the West fix the date in advance with calculations. Supporters say this removes confusion and the goal is simply to know when the month begins.

Each method has scholars behind it, and each is followed in good faith.

One moon for everyone, or each region on its own?

Scholars discuss whether one confirmed sighting applies to all Muslims worldwide. Some say yes. Others say each region follows its own sighting. A report from the time of the Companions is often discussed: a man saw the Ramadan moon in Syria, but when he reached Medina, Ibn Abbas said they had seen it a day later and would continue until they completed thirty days or saw the moon themselves (Sahih Muslim 1087). Some scholars take this as evidence for local sighting, while others read it differently. Because of this, the difference is one of scholarly judgment, not of right and wrong.

What should you do?

The simplest approach, especially for new Muslims, is to follow your local mosque or community and stay with it for the whole month, including Eid. Fasting with your community keeps things peaceful and avoids confusion. If your family overseas started a day earlier or later, that's fine too. See Why do Muslims celebrate Eid on different days? and How do I know which day Eid is?

Key points

Ramadan begins with the new moon, which can be seen on different days in different places.
Communities use local sighting, a sighting from another country or calculation.
Scholars have allowed more than one method, so a one-day difference is normal.
Follow your local mosque and stay with it all month.

Tip: Ask your mosque early which method it follows, and join its messaging group so you hear the announcement as soon as it's made.