The Brothers Karamazov is set in the tumultuous years following Russia's abolition of serfdom in 1861. (To be more precise, if the novel is set "thirteen years" before its publication, as the narrator tells us, this places the time of the novel around 1866.) By setting the novel in a provincial town rather than an urban setting, Dostoevsky can explore how the abolition of serfdom transforms social relations across different classes, from the peasants to the landed aristocracy.
The 1860s were a volatile period in Russian intellectual history. The abolition of serfdom was but one consequence of the influence of Western European philosophies of liberalism and socialism on Russian society. This period also witnessed the rise of radical political movements, including the People's Will, which assassinated Tsar Alexander II in 1881, the year Dostoevsky died. These ideas are embodied in intellectual characters such as Ivan and Rakitin.
But the novel also registers the strong conservative, Slavophilic currents that arose in reaction to these radical ideas. (Slavophilism is the celebration of the Slavic roots of the Russian people.) The novel is largely sympathetic in its portrayal of more conservative characters such as the elder Zosima, who stresses the centrality of the Russian Orthodox religion and champions the intrinsic moral superiority of the Russian people.
Although the novel is set in the 1860s, it registers the controversies over judicial reform in the late 1870s. The fact that Dmitri's jury consists of, to put it gently, men of mediocre intelligence is in line with the conservative view that juries cannot handle the demands of justice (source: Terras, Victor. A Karamazov Companion. Madison, WI: The University of Wisconsin Press, 1981. P 63).