Short answers, pulled from the story.
Modern dance is a broad genre of western concert or theatrical dance that primarily arose out of Europe and the United States in the late 19th and early 20th centuries. It developed as a rejection of, or rebellion against, classical ballet, and as a way to express social concerns such as socioeconomic and cultural factors.
Isadora Duncan, Maud Allan, and Loie Fuller pioneered new forms in the late 19th century in what is now called improvisational or free dance. Ruth St. Denis, Ted Shawn, Martha Graham, Doris Humphrey, Charles Weidman, Katherine Dunham, and Lester Horton later shaped distinctly American styles.
Modern dancers rejected ballet's strict movement vocabulary, the limited set of movements considered proper to ballet, and stopped wearing corsets and pointe shoes in their search for greater freedom of movement. Isadora Duncan considered ballet to be ugly and meaningless gymnastics.
Graham technique is the dance method developed by Martha Graham, hinging on the concepts of contraction and release. It focuses on the center of the body rather than ballet's emphasis on the limbs, coordinates breathing with movement, and emphasizes the dancer's relationship with the floor.
African American dance blended modern dance with African and Caribbean movement, including a flexible torso and spine, an articulated pelvis, isolation of the limbs, and polyrhythmic movement. Katherine Dunham, Pearl Primus, and Alvin Ailey were central figures, with Ailey creating his acclaimed work Revelations in 1960.
Postmodern dance developed in the 1960s in the United States and rejected the formalism of modern dance. It incorporated elements such as performance art, contact improvisation, release technique, and improvisation, and choreographers no longer created specific schools or styles.