Short answers, pulled from the story.
The NKVD, or People's Commissariat for Internal Affairs, was the interior ministry and secret police of the Soviet Union from 1934 to 1946. It succeeded the OGPU and held a monopoly on intelligence and state security functions. In 1946 it was renamed the Ministry of Internal Affairs (MVD).
In 1937 and 1938 alone, at least 1.3 million people were arrested and 681,692 were executed for what the Soviet state called crimes against the state. The Gulag population swelled by 685,201 during the same period, with at least 140,000 prisoners dying from malnutrition, exhaustion, and exposure.
The NKVD was led by Genrikh Yagoda from 1934 to 1936, Nikolai Yezhov from 1936 to 1938, Lavrentiy Beria from 1938 to 1945, and Sergei Kruglov in 1946. Each headed the agency during a distinct phase of Soviet internal repression.
Yes. Leon Trotsky, Stalin's most bitter international critic, was killed in Mexico City in 1940 by NKVD operatives. He is among the officially confirmed victims of the NKVD's foreign assassination program.
The Katyn massacre was the NKVD's execution of approximately 22,000 Polish POWs in 1940-1941. Chief NKVD executioner Vasily Blokhin personally oversaw and carried out thousands of the killings. On the 26th of November 2010, the Russian State Duma declared that the massacre was committed on direct orders from Stalin.
Sergey Korolev, who later became head designer of the Soviet rocket program and oversaw the first human space flight mission in 1961, was held in a sharashka. Aircraft designer Andrei Tupolev and writer Aleksandr Solzhenitsyn, who based his novel The First Circle on the experience, were also imprisoned in sharashkas.