To understand the history of Mormonism one has to understand the milieu of the early 19th century. As Joseph Smith, the founder, was coming of age the US an especially New York was undergoing a period of religious revival we now call the Second Great Awakening. The fervor got so pitched in Western New York it got the nickname "The Burnt Over District" because supposedly the fires of religious revival burned so strong there. Not surprising the area and time also produced the Seventh-Day Adventists, the Churches of Christ, the Methodist Holiness movement, and others. Plus a lot of smaller communities like the Oneida group. So that was the environment that Smith grew up in.

In that time and place a popular movement grew up called Campbellism after its early proponents Thomas and Alexander Campbell. They were kind of a Baptist offshoot, but didn't belong to the baptists, nor did they found their own church. Instead they preached wherever they could to whomever would listen. They preached the idea that the original Christian church had to be restored, they preached that spiritual gifts (prophesy, tongues, etc) would return, and they preached millennialism that Christ's return was eminent.

For whatever reason this clearly stuck with Smith so when he wrote (or translated) his history of Jews living in America, the theology in the book was almost a dead ringer for Campbellite teachings. And the Campbellites were primed to accept a prophet claiming to lead a restoration of the primitive Christian church. So when Smith appeared claiming to have translated an ancient scripture that matched Campbellite doctrines they jumped on board. Not all of them by any means, but a lot. In particular he caught and converted an important Campbellite pastor by the name of Sidney Rigdon. Rigdon had been the leader of the largest congregation (of any type) in Pittsburg and had just recently broken with the baptists and had moved to Ohio. So when Smith got Rigdon, he also got most of Rigdon's congregations, and he inherited a collection of traveling preachers. This gave the new church an early core of followers in New York, Pennsylvania and Ohio. Nine years later these enthusiastic early missionaries had expanded the church to nearly 18,000 members. By the time they migrated to Utah a decade after that, they had more than doubled again, and by 1854 there were nearly 70,000.

This early growth was fueled by similar movements in Europe and Mormon missionaries found tens of thousands of converts, especially UK and Scandinavia. But by the 1850s the greater religious revival movements were petering out, the Mormons had migrated to an somewhat inaccessible desert, and most importantly they had publicly announced the practice of polygamy. It had been practiced by the leadership in secret for 15-20 years before 1854, but it had always been publicly denounced. Between the general cooling of religious interest, the isolation, and the acceptance of polygamy, new conversions dropped off a cliff. In fact for a while the church contracted. It took he church nearly a decade to recover to its pre-announcement size.

For the next century or so the church grew largely through natural increase. Mormons are famous for large families for a reason after all. They always maintained large missionary forces, and the late 19th did see some areas of new growth. In particular the Pacific Islands saw a lot of new growth. Most of Polynesia is at least 10% LDS, American Samoa is 30% LDS, and Tonga is 20% LDS... partially explaining why BYU has so many Pacific Islanders on their football teams. But aside from that there weren't a lot of conversions.

By the early 20th century the church had banned polygamy and was trying to mainstream itself. They had become highly aware of their reputation as religious weirdos and were determined to do something about it. For example in WWI the church pushed enlistment for the first time once the Mexican-American war. This was a change, as the Mormons had largely sat out most of the 19th century wars and viewed the federal government with extreme suspicion.

In 1918 Heber Grant became prophet and he kicked the mainstreaming into overdrive. If there was a popular movement that was compatible, the Mormons were on board. They became prohibitionists. They removed anti-American speech from their hymns and temple rituals. They did everything they could to appear like "good neighbors." This didn't do much during Grant's life but it set the stage.

By the 1950s they had developed a reputation of being good folks, odd beliefs, but good folks. As the civil rights movement and other scares started hitting the press, the church started seeing its growth climb again. It was seen as an alternative to chaos, a good place to raise a family. By the 1960s the church was seeing percentage growth rates they hadn't seen since the early days, and crossed over the 2 million adherent mark. It was also the period where they really started to push missionary work outside the US for the first time in a long time. As sixties turned into the seventies growth in the US slowed. What had appeared stable and comforting in the sixties started looking less appealing. White shirts and ties were going out of fashion, and banning blacks was increasingly radical. The "good neighbors" growth was over.

But to replace the US conversions, Latin American conversions started booming. And in the eighties and nineties most new growth happened there. This stemmed from a religious revival that swept Latin America during those decades, especially in areas that transitioned to democracy. For example Brazil saw explosive growth from around 1985 to 1995 following the election of Tancredo Neves. Along with being new and interesting, the church was seen as distinctly "American" and drew in a lot of Ameriphiles across the region. As might be expected churches like the Jehovah's Witnesses and Seventh-Day Adventists also saw significant growth in the eighties and nineties.

Sometime in 1996, the church crossed 10,000,000 total members and about half that in regular attendees.

Unfortunately I can't talk about the more recent developments on church growth as they've had some fascinating developments over the past couple decades. But that's current events, not history.

To sum up the LDS church has seen three major periods of conversion growth. The first coincided with the Second Great Awakening, the second was as part of the counter-counter-culture in the sixties, and the third coincided with a Latin American version of the Great Awakening in Latin America.