Muslims will be surprised to learn than conversion to Islam was not easy (or encouraged) in the beginning of Islam. To convert to Islam meant to become an Arab; it involved being adopted by an Arab tribe. This is one reason why people formed an opposition group to the Islamic rulers. These people would become Shi’at ‘Ali or the Party of Ali bin Abi Talib. Ali and his followers believed conversion rules for non-Arabs should be made easy. But some people wanted the original rules to stand – if everyone converted to Islam, where would the jizyah tax come from? (Eventually, the lenient conversion rules would become normative for Islam.)
Once conversion became something encouraged, missionary work began in earnest. While the Islamic polity was spread by the sword, the religion was spread by wandering missionaries. Many people converted to enjoy the fruits of being an equal of their new rulers (and to be full citizen, rather than second-class citizens, and to avoid the jizyah tax); others converted out of liking the new religion. Many missionaries did a good job rephrasing Islam in terms the non-Muslims would understand, which in many cases introduced non-Muslim elements into Islam. (This is most prevalent in South Asia.)
Today, conversion is highly encouraged. Efforts in da’wah (literally, “invitation,” now usually referring to missionary work) are encouraged by all Muslims. Indeed, some Muslims have said that Muslims living in non-Muslim lands are living there against Islamic law unless they engage in da’wah. Whether it’s educating people about what they want people to think about Islam, or leading people to conversation, ordinary Muslims lead many people to conversion. There are missionary organizations – Tablighi Jamaat and Jamaat-e Islami are two – but many focus on “converting” Muslims to true Islam or to train Muslims in missionary methods.