The Borzoi became popular in the period of time before the Russian Revolution in 1917. This period of time in Russian history is commonly referred to as "Tsarist Russia" because it was when the Tsars were in power. The Russian Tsars were immensely important to the Borzoi for many reasons: they are, in fact, the very cause of the popularity of the Borzoi as well as being the responsible parties for the breeding of the Borzoi into the dog we know and love today.
Before the 1917, a person could only own a Borzoi if it had been given to them by the Tsar as a gift. This, of course, limited the ownership of the dogs to the royalty and nobility. Borzoi were not owned by commoners until after the revolution.
In Tsarist Russia hunting trials were the popular way for the nobility to spend their time. These trials were ceremonial outings that lasted several days and included several mounted hunters (always members of the nobility or royalty) and multitudes of Borzoi. The Borzoi would often chase down small game for their masters, but sometimes the dogs would be sent after wolves. The dog's job would be to catch and hold the animal by the throat until the human hunter came and killed it. These hunting trials were set up like tests for the breed in order to determine which dogs were the fastest and the most intelligent. Only those dogs would be bred to strengthen the Borzoi line.
The most famous breeder of the Borzoi was Russia's Grand Duke Nicholas Nicolaievich. He is the nobleman who took the breeding of the Borzoi to new levels. Before the Grand Duke, Borzoi were commonly one of five major types: the Stepnaya, the Hortaya, the Tazi, the Krimskaya, and the Hort. The Hortaya closely resembled the short-haired version of today's Borzoi. Nicolaievich's breeding brought about today's long-haired breed standard of the Borzoi, called the Psovaya Borzaya, or Psovoi.
During the Russian Revolution in 1917, Tsars and nobility were attacked and their possessions destroyed. Dogs were not exempt from this, especially the Borzoi, who had become symbols of aristocracy. The dogs were often tortured and killed, leading many nobles to shoot their dogs themselves so the dog would not have to suffer torture at the hands of soldiers.
Of course, the Borzoi breed did survive. In the 1940s, a soldier named Constantin Esmont discovered large numbers of Borzoi in the Cossack villages in the Russian countryside. Thanks to Esmont, the breeding of the Borzoi was regulated by the Soviet government to avoid over-breeding and irresponsible breeding of the dogs. This regulation led to the two basic kinds of Borzoi we know today: the short-haired Borzoi still used as hunting dogs in Russia, and the long-haired Borzoi who is a companion and dog show champion.