What did Baudelaire mean when he wrote that modernity refers to the ephemeral, the fugitive, or the contingent?

In addition to being a great poet, Baudelaire was renowned as a perceptive, insightful art critic. To this day, many regard his evaluation of Delacroix as unsurpassed. At the time when Baudelaire wrote "The Painter of Modern Life," the French art world was experiencing convulsive, revolutionary change. Baudelaire's article coincided with the notorious Salon des Réfusés, an exhibition of the works of exciting young artists such as Manet, whose work had been rejected by the conservative Paris Salon.
The work of such artists was scandalous to the French art establishment, both morally and aesthetically. The prevailing academic standard was ultra-realistic and dealt with elevated themes from history and antiquity. Manet's Le déjeuner sur l'herbe, with its depiction of a nude woman picnicking with two fully clothed men, departed so completely from the established rules, in terms of both its subject matter and how it was painted, that it was considered the epitome of the new style by both supporters and detractors alike.
For his part, Baudelaire was an enthusiastic champion of modern art. He saw works such as Guy’s and Manet’s as capturing the essence of modern life. That life was characterized by its endless hustle and bustle, its ceaseless flux. Everything in this world is ephemeral—here today, gone tomorrow. Moreover, it is radically contingent—that is to say, what exists in this rapidly developing society might well not have existed at all. And modern life is fugitive—it has no home of its own, transcending as it does societies and geographical boundaries. In the face of such a world, all the artist can do is capture those fleeting moments that careen in and out of consciousness with such alarming rapidity.
For Baudelaire, Guy is the paradigm example of the modern artist. He is possessed of the remarkable ability to paint

"the passing moment and all of the suggestions it contains."

This is the crux of the matter for Baudelaire. Eternity is to be found hidden in the fleeting, passing moments of modern life. And it is the heavy responsibility of the modern artist to set it free—to illuminate the eternal in the temporal, the transcendent in the imminent, the permanent in the temporary.

The full quotation to which you are referring is taken from Charles Baudelaire's "The Painter of Modern Life." It reads:

By ‘modernity’ I mean the ephemeral, the fugitive, the contingent, the half of art whose other half is the eternal and immutable…

The statement describes the work of Constantin Guys, who is referenced in Baudelaire's essay simply as "Monsieur G." Guys (December 3, 1802 – December 13, 1892) was a journalist and illustrator best known for his reporting on the Crimean War. The essay, while overtly focusing on the work of Guys, also defends the work of Impressionist and Symbolist painters that Baudelaire admired.
In this essay, Baudelaire is arguing that "modern" art must move behind the classical ideals of grandeur and timelessness to capture the fleeting impressions of everyday life, even at its most squalid. Unlike the Naturalists, he is not arguing for realism so much as artistic vision, which he says is transformative, creating beauty from fleeting impressions. He is also arguing for casting off the straight jacket of artistic tradition and embracing the modern world, rather than imitating ancient models. 
Guys' work, in its simple technique and response to the details of immediate experience, represents Baudelaire's ideal of the flâneur, strolling through the city and observing its varied inhabitants.