Princess R. was Pavel Petrovich's enigmatic love interest, the woman for whom he ruined his career. She was a well-respected society woman with a husband and children, but she was also known for being very eccentric and constantly moving back and forth between Russia and Western Europe. Less known was that "at night she wept and prayed, finding no peace in anything" (7.2).
Pavel Petrovich wooed her easily, but his obsession only grew with time. Her behavior was "a maze of inconsistencies" that fascinated him (7.2). After some time passed, he gave her a ring with a sphinx engraved on it, suggesting that she herself was as mysterious as the sphinx.
When Princess R. fled Russia, Pavel Petrovich resigned his post in the military to follow her. He eventually caught up with her in Baden, but she fled again and after that he essentially gave up and "undertook nothing new" (7.6).
Later, Pavel Petrovich heard that Princess R. had died in Paris "in a state bordering on insanity" (7.6). He received a package from her a few weeks later. In it was his ring; she had etched a cross over the image of the sphinx. Her note to him said only that the solution to the riddle was the cross.
Princess R. is a minor figure in the novel, but her relationship with Pavel Petrovich is, in many ways, echoed by Bazarov's relations with Anna Sergeyevna.