Background: The Mexican Revolution began in 1910, in response to Porfirio Diaz's regime, which was essentially a dictatorship. Right before the Revolution, he allowed "free" elections and then jailed his major rival. After the Revolution gained momentum, Diaz abdicated and fled to Europe.
The immediate effect of this was to sweep Francisco Madero into office, who had earlier challenged Diaz and tried to run a fair presidential campaign, leading to him being harassed and self-exile, where he proclaimed the Revolution.
The revolutionaries, on the other hand, were unsure of what they had fought for. The first schism occurred during the first phase of the Revolution, when Madero was winning against Diaz. His cabinet included many family members, and later Diaz supporters. Madero's generals, and later the population, grew frustrated as his extremely slow pace of reform.
Eventually, several new movements against Madero began, including a counterrevolutionary one lead by Diaz's son and a loyalist general, who met up and stormed the capital. Huerta, who was in charge of the defense of the capital, defected and overthrew Madero, setting up his own dictatorship.
After even more revolution, and intervention by the United States (in the form of them occupying Veracruz), which had threatened to unite the various revolutionary factions against the external threat, Huerta fled the country in mid-July 1914, leaving the presidency to Carranza.
Carranza was more of a moderate than Zapata, Villa, or their supporters. They felt he didn't live up to the ideals of the Revolution (which to them was chiefly agrarian reform, as an immense percentage of the poor population worked on vast plantations owned by the wealthy), and it showed when the Constitution of 1917 was drafted; the constitution was heavily inspired by the Villista and Zapatista factions. Carranza more or less ignored a large portion of it, leading to yet another stage in the Revolution. The U.S. more or less sided with Carranza, and Villa, who had worked hard to get into the U.S.'s good graces, was outraged. In response, he crossed the border, robbed a train, and captured a small town, pretty much burning it to the ground.
In response, the U.S. sent John J. Pershing on a punitive expedition, sending around 5000 men to capture or kill Villa. Over the course of a year, they couldn't track him down and were forced to head back to the U.S. when it came close to war.
However, Villa was eventually defeated by machine guns, which over the course of two battles decimated his cavalry heavy army. In the south, Zapata was lured into a trap and assassinated.
Carranza was overthrown in 1920, by several officers including Álvaro Obregón (who was the next in office), who didn't want the nation to revert to civilian control. Under Obregón, Mexico was peaceful for a while. A few years later, another revolt sprang up, over Obregón's nomination of his successor and a move away from older policies.
In 1926/1927, after the Mexican Revolution proper, but near enough to warrant mention, another civil war erupted; this time over matters of religion. The post-Revolution government was highly secular, and reversed decades of favoring the Roman Catholic church in exchange for heavy regulation and anticlerical policies. Many heavily disagreed with the policies, and before long there were shootouts between federal troops and the Cristeros, as they called themselves. The rebellion was eventually put down, but at heavy cost. Afterwards, the anticlerical laws remained, but they were rarely enforced.