Pentecostalism takes its name from the event at Pentecost as described in Acts 2. In that passage, there was coming of the Holy Spirit that was accompanied by speaking in tongues. Pentecostals identify with that experience and seek it for their own life.
The origin of Pentecostalism is often traced back to Azusa Street Revival in Los Angelas under the preaching of William Seymour (1906-1915). While that was a significant event, it goes back farther.
Pentecostalism emerged partially out of Methodism. Wesleyan theology included a second act of grace called sanctification. It was believed that a Christian could experience full sanctification in this life.
A number of holiness movements emerged in the 19th century. These heavily emphasized the present work of the Holy Spirit. There was an increasing focus on the miraculous.
A.B. Simpson (1843-1919), founder of the Christian and Missionary Alliance, brought miraculous healing back as a major focus of Christian experience. Around the turn of the century, there were numerous claims of speaking in tongues.
The beginning of the twentieth century saw an explosion of Pentecostal groups. They were marked by diversity. Racial groups worshiped together and women had important places in leadership. Unfortunately, these progressive movements diminished over time as the Pentecostals became more organized and emulated other fundamentalist groups.
Part of the diversity was with regard to theology. There were mainly three streams of Pentecostals: 1) holiness Pentecostals, who believed in both the baptism of the Spirit and sanctification, 2) oneness Pentecostals, who rejected the Trinity and saw Jesus as the one and only God and 3) classical Pentecostals, who accepted both the Trinity and gradual but not full sanctification.
The defining theology of Pentecostals, then and now, was that there was an experience called the baptism of the Holy Spirit that was available post-conversion for all Christians. The evidence of this baptism was that of speaking in tongues. Speaking in tongues was not a foreign language but ecstatic speech.
In addition to speaking in tongues, there is also an emphasis on miraculous healing and eschatology. Eschatology is often of the pre-tribulational, pre-millennial form.
It should be noted that Pentecostalism and the prosperity gospel are not synonymous. Some Pentecostals do embrace the prosperity gospel and some prosperity proponents would not identify as Pentecostals.
One study suggests that Pentecostalism represents 26 percent of world Christianity, second only to Roman Catholicism.