Short answers, pulled from the story.
Emo originated in the Washington, D.C., hardcore punk scene in the mid-1980s, with Rites of Spring forming in 1983 as one of the pioneering bands. The style spread nationally through homemade zines and vinyl records after most of D.C.'s founding emo bands had broken up by 1986.
Brian Baker of Minor Threat and Dag Nasty coined the phrase "emotional hardcore" during emo's formative years, using it as a put-down. The term was then adopted by Thrasher magazine and spread through the D.C. punk scene, eventually shortening to "emo", though its exact coinage remains disputed.
Jimmy Eat World's 2001 album Bleed American went platinum and opened the door to mainstream success. My Chemical Romance, Fall Out Boy, and Panic! at the Disco followed, with Fall Out Boy's From Under the Cork Tree selling 2,700,000 copies in the United States and Panic! at the Disco's A Fever You Can't Sweat Out certified quadruple platinum by the RIAA.
Many artists reject the label because it became a stigma. Guy Picciotto of Rites of Spring called it "retarded", Gerard Way of My Chemical Romance called emo "a pile of shit" in 2007, and Bayside vocalist Anthony Raneri said emo became "a dirty word" when hipsters used it to demean rock artists they considered less credible than indie groups like the Strokes.
Screamo is a dissonant offshoot of emo that developed in San Diego in 1991, beginning at the Ché Café with bands such as Heroin and Antioch Arrow. It is characterized by short songs, chaotic execution, screaming vocals, and influences from Washington, D.C. post-hardcore, straight edge, and gothic rock bands like Bauhaus, setting it apart from emo's more melodic and confessional approach.
The emo revival, or fourth wave, began in the late 2000s as a reaction against the commercial third wave, drawing on the sound of 1990s Midwest emo. Pennsylvania-based groups Tigers Jaw, Glocca Morra, Snowing, and Algernon Cadwallader were among the earliest bands; a 2018 Stereogum article cited Algernon Cadwallader's 2008 LP Some Kind Of Cadwallader as the revival's watershed release.