Dick Grayson is the Robin who died before Batman v Superman: Dawn of Justice, and here's what is known about how it happened in Zack Snyder's plan for the DC Extended Universe. Among all the moments in Batman v Superman that fans obsess over, the death of Robin is one that fascinated many ever since it was teased in the film's marketing.
Robin's death was a signature comic event in 1988 and made the passing of Jason Todd, the second Robin, a huge moment. Due to Jason's ties to being killed (and then returning as Red Hood), many fans assumed that he was the Robin who died in Snyder's story, with Warner Bros. even saying Jason was the Robin who died. In the years since Batman v Superman hit theaters though, Snyder has confirmed that Jason wasn't the one who died, it was Batman's original Robin, Dick Grayson. The loss of a protege still haunts Batman, and even though the movies haven't focused on Dick's death, there is a decent amount of information known about how he died.
The clearest detail about Dick Grayson's death in the DCEU is that it came at the hands of Joker. Snyder has revealed in past interviews that he believes Dick died about ten years before Batman v Superman, which helps add some context for the moment in the DCEU timeline. Snyder even described Joker as young at the time of Robin's death, so it could've come early on in Joker and Batman's careers. Joker wanted Batman to make no mistake about who was responsible for the death too, thanks to the particular paint job he gave the suit. But, it isn't yet known what Joker did to kill the Boy Wonder. It was later revealed in Suicide Squad that Harley Quinn was an accomplice in Dick's death, although that doesn't match up with the DCEU timeline.
Following the death of Robin, the loss had a tremendous impact on Batman. Dick's death is what took Batman down a darker path at first, resulting in the version of the Dark Knight viewers met in Batman v Superman. Before the film, though, Batman made sure to include Robin's suit (and bike) in the Batcave to continually remind him of his failure and motivate him not to let that happen again. Batman did exact some revenge on Joker for the death of Robin, though, as he punched out all of the Clown Prince of Crime's teeth later on.
Snyder's plans for the DCEU were cut short due to Justice League, but more information on Dick Grayson's death could still be on the way. The director has said that his version of Justice League included some more details on Robin's passing and the effect it had on Batman, so that could be included in the Snyder Cut of the film now. Whatever else Snyder reveals about Dick, there were no plans to resurrect the character in future films. Instead, he has said that his intention was for Carrie Kelly to become Batman's next Robin. This could mean that Dick, Jason Todd, and maybe even Barbara Gordon had little to no roles in Snyder's DCEU plans.