The Nineteenth Century
American painting was virtually absent from the international scene.  American artists drew largely from the work of European masters in creating their art.
A few American figures stand out from this era:

Thomas Eakins (1844-1916)
Considered by some the premier American painter of all time; widely regarded as the best of the nineteenth century
An early realist; an American scene painter
His art drawn from American life
Attacked while teaching at the Pennsylvania Academy of Fine Arts in 1870s for his insistence on using nude models
His portraiture compared to Rembrandt's
Known for his scientific study of the human form, human motion

See "Max Schmitt in a Single Scull," 1871
and "Miss Van Buren," ca. 1886

compare to
Rembrandt's "Portrait of a Lady with an Ostrich-Feather Fan," ca. 1660
 

Eakins's masterpiece, "The Gross Clinic," 1875

Also noteworthy from this era is
Winslow Homer (1836-1910)
Also a realist, naturalist
Almost completely self-taught (an autodidact)
 

Homer's "High Cliff, Coast of Maine," 1894
"I regret very much that I have painted a picture that requires any description."
-- Homer, when asked for a few descriptive lines about "The Gulf Stream"

James McNeill Whistler (1834-1903)
referred to his works as "ideal constructions"

"Arrangement in Gray and Black Number I: Portrait of the Artist's Mother," 1871, almost always referred to as "Whistler's Mother." Hangs in the Louvre.
See more of Whistler's works here; read a brief bio here.

Mary Cassatt (1844-1926)
An impressionist painter
Invited to exhibit by Edgar Degas
Known for scenes of family, children (q.v. "The Cup of Tea," a portrait of her sister, ca. 1879)
Awarded the Legion of Honor in 1904