Modified from an earlier answer

Much of the popular memory of Catherine's promiscuity is a gross exaggeration built on small kernels of truth. Catherine did have a series of lovers and sometimes used sex early in her reign to secure her position. For example, Grigory Orlov led the coup that overthrew Catherine's husband, Tsar Peter III and was one of Catherine's lovers. It should be noted too that Peter III had done much to alienate the Romanov court, so Catherine's coup was not unsurprising even without the romance backstage. Catherine's lovers like Potemkin also became her personal agents throughout the empire. In addition to sex, she would employ all sorts of stratagems to tie them together with bonds of affection and loyalty such as hand-sewing herb-filled pillows for them when they traveled. These men were not from the upper echelons of Russian elite society and Catherine's patronage represented a significant improvement in their social status. Thus the term "Catherine's lover" must be understood as something greater than just a sexual relationship.

When dealing with Catherine the Great, it's very hard to parse out what what might be construed as her sexual or romantic appetite from her position as empress. Being a female ruler in a vast empire like Russia necessitated a degree of dependency on her male servitors and she was very canny about choosing able ones. Her contemporaries witnessed this and thus sprang legends about her sexual proclivities. It did not help matters that her Prussian neighbor, Frederick the Great, had a strong misogynistic streak even by the standards of the day; Frederick's homosocial court and the absence of women in his entourage was atypical for a monarch. When Catherine ascended the throne after the coup, Frederick wrote to his brother in October 1774 that "a woman is always a woman and, in feminine government, the cunt has more influence than a firm policy guided by straight reason." Frederick did not really discourage rumors about his rival from being spread in Prussia. This is where popular stories like the horse came in, it served to bring low a woman who had against expectations strengthened the Russian imperial state.

The French Revolution expanded these preexisting rumors of Catherine's depraved appetites. John T. Alexander's biography of Catherine posits that the origin of the horse story is the 1791 Ochakov crisis in which the French and British press visualized Catherine's seizure of Ottoman territory in overtly sexual terms, such as this famous print. Russia's support of counter-revolution (however tepid) and the lifting of censorship against monarchical figures meant that Catherine became a fodder for broadsheets. These tales of debauchery dovetailed with both anti-Bourbon broadsheets, which also regaled their readers with tales of unnatural lusts, and the wider demi-orientalism in France towards Russia which portrayed Russia as a semi-Asiatic land governed by tyrants.

Although the reality of Catherine the Great and her lovers is less salacious than popular memory holds it, the fact that this mythology has become so entrenched reflects the complex nature of reactions to women in power, especially when one was as successful as Catherine.

Sources

Alexander, John T. Catherine the Great: Life and Legend. New York: Oxford University Press, 1989.

Madariaga, Isabel de. Catherine the Great: A Short History. New Haven: Yale University Press, 1990.

Wortman, Richard. Scenarios of Power: Myth and Ceremony in Russian Monarchy. Princeton, N.J.: Princeton University Press, 1995.