In these four years, millions of people were arrested and killed, either by execution (sometimes torture) in prison, or by overwork and malnutrition in the labour camps, or execution there. Hardly a family was left untouched, especially in the western and central USSR.

Why did Stalin launch the terror and carry it to such extremes?

The most likely answer is that he saw any opposition, real or potential, as a deadly threat to himself and that this perception confirmed his determination to hold absolute power.
At the same time, like other Bolsheviks, including Lenin, Stalin believed that terror was a legitimate political weapon.
The difference was that while Lenin and other Bolshevik leaders opposed the use of terror against their own colleagues, Stalin had no such reservations.
Finally, we know that Stalin believed terror to be the most effective means of making people obey and work hard.

It seems that the impulse for launching the great terror was criticism of Stalin's policies and methods within the party leadership in the years 1930-32, i.e. the period of forced collectivisation.

At that time, the most significant opposition came from Ryutin. Ryutin was expelled from the Party and arrested in September 1930, but the OGPU (Security Police) acquitted him of any criminal intent and he was only given a warning.
However, in 1932, he and a group of minor party officials - some of whom were followers of Bukharin, who had opposed collectivisation - wrote "An Appeal to All Members of the All Union Communist Party (Bolshevik)." This paper, known as Ryutin Platform, proposed an economic retreat, that is, a reduction of investment in heavy industry and the liberation of the peasants, allowing them to leave the collective and state farms.
The authors condemned Stalin as "the evil genius of the Russian Revolution." They pointed to the lawlessness and terror existing both in the party and in the countryside, to the collapse of genuine planning, and said the press was reduced "in the hands of Stalin and his clique to a monstrous factory of lies."
Finally, the appeal stated: "Stalin and his clique will not and cannot voluntarily give up their position, so they must be removed by force."

Stalin seems to have interpreted this as a call for his assassination, but the Politburo refused his proposal that Ryutin be shot. The party leaders still opposed the death penalty for one of their own. Ryutin and his supporters were, however, expelled from the party. He received a ten year sentence and later died in prison.

It was also clear at the 17th Party Congress, held in January-February 1934, that many deputies wanted a relaxation of the collectivisation drive and that Kirov was a very popular figure. Stalin must have decided to get rid of his critics and potential rivals, but he needed a pretext.

Stalin's pretext for the purges in the party, which developed into the mass terror, was the assassination of Kirov on December 1, 1934.
Kirov was widely regarded as Stalin's heir apparent and was popular in party circles. Kirov opposed Stalin's brutal methods of collectivisation, and received many more votes than Stalin for reselection to the Central Committee (and thus election as Secretary General) at the 17th Party Congress of 1934. In fact, only three votes seem to have been cast against Kirov, while some 270 were cast against Stalin. However, Stalin's henchmen are said to have destroyed these except for three, also leaving three votes against Kirov.

Finally, though Kirov refused to run against Stalin for the post of General Secretary, and told him so, Stalin apparently concluded that Kirov was a deadly threat to him.

It seems that NKVD operatives, under Stalin's orders, used Leonid Nikolaev, a party member known for his disturbed mind, to kill Kirov in his own office building in Leningrad. Later Nikolaev, as well as all others involved, including Yagoda, were killed off in one way or another.

Stalin then raised a great hue and cry claiming the whole party was in danger, having been "penetrated" by spies and foreign agents. Thus, Kirov's murder was Stalin's pretext to start a series of purges in the party.

There were mass arrests, which included not only the suspects, but also their families, supporters, friends and acquaintances. It is estimated that some 40 million Soviet citizens lost their lives under Stalin's rule (plus 28m more if you count those killed in WW2)

The visible part of the purges were the show trials of well known Bolshevik leaders, all of whom had opposed Stalin at various times in the past. Although all were accused of belonging to "Trotskyite" conspiracies, and some of spying for foreign powers, almost all were also accused of sharing the "Ryutin Platform."

These trials were:
(1) The Trial of the 16, August 1936, when Kamenev, Zinoviev, I. N. Smirnov, G. E. Yevdakimov and others, were accused of being part of a "Trotskiite-Zinovievite Terrorist Centre," and of organising a "terrorist plot" against Stalin and his supporters. The accused were forced to implicate Bukharin, Rykov, and Tomsky (the last committed suicide on being implicated). The accused had been promised their lives and safety for their families, if they "confessed," but they were shot the day after their conviction.
(2) The Trial of the Anti-Soviet Trotskiite Centre, January 1937, in which Trotsky, who was in exile since 1929, was the arch villain. This time the accused were headed by . Pyatakov, who had consistently supported Trotsky in his disputes with Lenin and Stalin. The other accused included the prominent expert on foreign affairs and former leading member of the Trotskiite opposition, Karl Radek, Grigorii Y Sokolnikov, a diplomat and member of the "Left Opposition" (Trotsky); Serebryakov, a leading member of Trotsky's former group, and thirteen others. They were forced to implicate Bukharin, Rykov, and Tukhachevsky. Some were executed and some died in labour camps.
(3) The Trial of the 21, March 1938, in which the key defendants were Bukharin, Rykov , a leader of the "Right Opposition" against collectivisation; Krestinskii , who had been the Soviet ambassador in Berlin in 1922-30, and Yagoda, the NKVD chief who had conducted the inquiry into the assassination of Kirov and organised the purges.
Again, most were executed, while others died in the camps.

Trotsky, who was in the West, publicly denied the charges.

Also in those years, Polish communist leaders were duly arrested and executed, or died in the camps. Finally, the Polish Communist Party was dissolved in 1938, on the charge that it had been infiltrated by the Polish police .

In the period of the Nazi-Soviet Pact (August 23 1939 -June 22 1941), Stalin even delivered some German communists to the Nazis as a token of his good will.

We should note that the purge trials were only the tip of the iceberg. In all, some 90% of the delegates of the 17th Party Congress of 1934 were purged, and most of them were killed.

Moreover, aside from the victims' families, friends, and dependents, the NKVD had regional quotas to fill, so charges were fabricated to fill them. The NKVD investigators were themselves under the gun. If they did not produce the required number of confessions/convictions, they were arrested and sent to labour camps.

Of course, Stalin's enemy no. 1 was Trotsky. He was murdered by a Stalinist agent in Mexico in 1940. The assassin, who was helped by certain NKVD operatives to worm his way into Trotsky's confidence killed Trotsky in his study by driving an ice pick through his brain.

The Purge of the Soviet Military.
The Army. Navy, and Air Force Officer Corps was decimated..

As a result of the military purges, four out of five Soviet marshals, some 90% of the generals, 80% of the colonels, and in all, 80% of officers above the rank of captain, were shot or put in labour camps.

This greatly weakened the army.
You might wonder why Stalin did this if, as seems likely, he feared invasion by Germany, Britain or France.
It seems likely that Stalin feared the mounting criticism of his policy in both party and army circles, and viewed some popular army leaders as a serious threat to himself.
Finally, he disagreed with the army leaders' views on what the army should be, supporting the idea of a people's army rather than a professional army.