A subpoena is a document issued by a court to a witness, who must then testify or face punishment.
On June 13, 1931, Emmett Brown was supposed to deliver a subpoena to Arthur McFly. However, he never got around to doing so. This did not affect matters in the long run, as Kid Tannen, the person that Arthur was meant to testify against, was still captured and imprisoned on August 25 of that year.
When Marty McFly traveled back to 1931 to try and break Emmett's older self out of jail, he needed to enlist the younger Emmett's help in building a rocket-powered drill. Emmett could not comply, as he needed to deliver the subpoena to Arthur before he could help Marty. Arthur was refusing to leave his apartment, as Kid had ordered him to stay there unless told otherwise by him. Marty recorded the voice of an angry Kid Tannen, and playing it back underneath the window of Arthur's apartment tricked him into coming down and receiving the subpoena. As a result of this, Arthur testified against Tannen at the courthouse and was targeted by Tannen's men, who knocked him unconscious, took him to the secret basement under the Sisters of Mercy Soup Kitchen for interrogation, and murdered him.
This change was felt by Marty later that night, who began to vanish from existence due to his grandfather's premature death. He traveled back in time and accessed the soup kitchen, where he was able to incapacitate Arthur's kidnappers and take Arthur to safety in the DeLorean time machine. Dropping Arthur off outside Hill Valley, they told him to lie low and await contact from a person called Sylvia. Unknown to Arthur, he had already met Sylvia, under her stage name Trixie Trotter. Knowing that Arthur would later marry Sylvia and father George McFly, Doc and Marty surmised that the timeline was repaired and traveled back to 1986.