The Superman Symbol is the emblem shaped like an "S" that Superman and most of members of the Superman Family (Supergirl, Superboy, Steel...) -and of the House of El in several continuities- wear on their chests in different versions of their costumes throughout the years. Its first appearance was in Action Comics #1 as part of the character's original design by Jerry Siegel and Joe Shuster. There have been multiple origins presented for this symbol.
Prior to the Crisis on Infinite Earths, a young Clark Kent chose the S-Emblem when he put together his costume among other reasons because it stood for "Superboy"., although later it would be told Jonathan Kent designed the "S" and his wife sewed the costume. When Kara Zor-El's parents resolved to send their daughter to Earth, Allura sewed her costume, adding the S emblem so Superman would know Kara wasn't an enemy.
Superman's origin story has been rewritten and rebooted many times over the years; most notably in John Byrne's 1986 The Man of Steel, Mark Waid's 2003 Superman: Birthright, Geoff Johns' 2009 Superman: Secret Origin and Grant Morrison's 2011 Action Comics. There are several correct definitive historical versions each valid at a different time in his career as a result.
John Byrne's 1986 mini-series The Man of Steel had Clark Kent and Jonathan Kent design it together independently while they were trying to create his costume. Later heroes adopted the symbol to show they were inspired by Superman.
Mark Waid's 2003 reboot Superman: Birthright established the symbol as Kryptonian. His parents Jor-El and Lara sent him to Earth with a blanket featuring the symbol. Clark originally believed that it was his family crest, but witnessed greater cultural significance surrounding it in his historical texts and came to understand that it was a symbol of hope.
Geoff Johns's reboot Superman: Secret Origin cemented the Kryptonian origin of the symbol. Upon watching a holographical recording of Jor-El wearing the symbol, Martha Kent guessed it was some sort of family crest and sewed it on her son's Superboy outfit. Supergirl's arrival and her parents' rescue confirmed it was the House Of El's family crest.