This varies to where you look at usually in the modern day Muslim world Islam plays a huge role in society, but you are asking what role Islam played in surrounding societies who had not accepted Islam. And example would be the island of Malta. This island was under Arab rule for a long time as a result, the modern Maltese Language is a descendent of Sicilian Arabic. In Islamic spain, you are still able to witness the remnants of the Islamic culture that once flourished there. Another example would be traditional African religions, although the land that they flourished on did not convert to Islam, due to trade with neighboring principalities that had converted to Islam, Islamic practices were often mixed with local traditions.