According to what I have read, the 1807 Insurrection Act allows the president to call the United States Armed Forces and the National Guard:
When requested by a state's legislature, or governor if the legislature cannot be convened, to address an insurrection against that state (§ 251)
To address an insurrection, in any state, which makes it impracticable to enforce the law (§ 252)
To address an insurrection, domestic violence, unlawful combination or conspiracy, in any state, which results in the deprivation of Constitutionally-secured rights, and where the state is unable, fails, or refuses to protect said rights (§ 253)
Usually, these criteria have to be met to use the act, like when Ulysses S. Grant used it on October 1871 to suppress the Ku Klux Klan or when Thomas Jefferson first used it in 1808 to detain those violating the Embargo Act. Also, the Act as been modified twice:
1861: the federal government can use the National Guard against the will of the state government in the case of "rebellion against the authority of the government of the United States", implemented for the Civil War and unrest that came after.
1871: section 253 edited to fight against insurrection from the Ku Klux Klan.
So, the process for invoking the act seems to be for the president to claim such action is necessary because it fits into one of the three requirements. The act was created to be implemented during acts of insurrection when the National Guard may not be sufficient or available to use (ex: people going against embargoes to support enemy nations in violent ways, Civil War where opposing states and their allies may cause civil unrest & attempt to use the National Guard as an insurrectionist force, large scale terrorist groups like the Ku Klux Klan post-Civil War, etc.)
- While I am not sure about the modern status of the KKK, at the time the Klan was called a terrorist organization in 1870 by a federal grand jury. As someone pointed out in the comments, this was before the United States called certain organizations terrorist groups, so they were called a group that "instilled terror". However, as the American Experience from PBS points out, they still essentially were considered a terrorist group by our standards and if the Insurrection Act was implemented today, it might be necessary for the President to mention that whatever group he is opposing was complacent in domestic terrorism:
In 1871 Congress also passed the Ku Klux Klan Act, which allowed the government to act against terrorist organizations. Grant did not rigorously enforce these laws, although he did order the arrest of hundreds of Klan members. But with the over`whelming support of the Klan in the South, convictions proved difficult to obtain, and the financial panic of 1873 would distract the North from the problems of Southern racism. -PBS, American Experience (http://www.pbs.org/wgbh/americanexperience/features/grant-panic/)