Indonesia has more Muslims than any other country. However, within that community there are differences.
When many parts of Indonesia converted to Islam, they kept older beliefs, and their religion became a mixture of Islam with other ideas and practices. Some of this is due to the missionary efforts of men like Sunan Kalijogo, one of the Wali Songo or Nine Walis, who combined Islamic ideas with traditional rituals such as the Garebeg.
Because much of Indonesia was far from the rest of the Islamic community, most of the new Muslims of the 1500s had no idea that their "Islam" was different from the regular Islam of Egypt, Arabia, or Turkey. In those days, travel was dangerous, long and expensive, and only a few people were able to make the hajj to Mecca.
By the late 1800s, new technology was changing the situation. Steamship travel allowed many more Indonesians to make the hajj, or even study their religion in Cairo or Mecca. Newer and cheaper printing methods allowed more people to read Islamic texts.
Some Indonesians decided to change their way of religion to avoid mixed practices, and to follow Islam the way that the rest of the world did. These people are sometimes called "modernists", although they often call for a return to old traditions. People who choose to stick to the way Islam is practiced in places like rural Java are sometimes called "traditionalists".
In Indonesian history, these differences have come to the surface from time to time. The "Padri War" in the Minangkabau region of Sumatra during the 1820s and 1830s was partly a conflict between "reformers" who wanted a more standard Islam and the traditional leaders who did not want to change. In Aceh in 1945, the traditional leaders were viewed as being pro-Dutch, and were swept away by pro-independence feelings as well as the idea of Islamic reform. A more destructive force was the Darul Islam movement under Kartosuwirjo, which fought against the new Republic of Indonesia. This sort of extremism is not widely respected.
Today there are two large, respectable organizations for Muslims in Indonesia: Muhammadiyah which tends to be more "modernist", and Nahdlatul Ulama which tends to be more "traditionalist".
In the 1950s and after, mystical books were published on Java such as "Darmogandul" and "Gatoloco", which were Islamic in name but not in substance.