Dick grew up in the circus with his parents (Mary and John), and was an acrobat with them, the "Flying Graysons". When he was 8 and performing in Gotham, his parent's line was sabatoged by Tony Zucco, and they fell to their death. Dick was taken and put in a juvenile detention center (since social serves was full), but was later sent to an orphanage. He realized his parents were murdered, and snuck out at night, trying to avenge them. He was found by Batman, and adopted by Bruce Wayne. Batman helped him realize that killing Zucco would be wrong, and after training him, they put Zucco in jail. Dick created the Robin mantle, named after his mother's nickname for him (he was born on the first day of spring), with his family's colors. Dick started when he was 9, making him the first Robin, and the youngest sidekick.
As the first Robin, Dick is in a lot of comics and shows, and sometimes he seems to be in two places at once. As best as I can tell, he continues with Batman for a while, but also joins the Young Justice (TV series) team, a covert team for the Justice League. He also forms the first Team Titans group and fights crime with them in Jump City. Major events during his time as Robin include being severely wounded by Two-Face, and shot by the Joker. After both, he and Batman fight, and he's "fired" from being Robin. During his Teen Titan days, he's also briefly coerced into working as Deathstroke's (a well known mercenary) apprentice. Deathstroke put hidden bombs in his team's blood, but Dick managed to escape, and I'm uncertain if Batman ever found out about it.
After leaving the house (after being fired), Dick starts as a new vigilante in Bludhaven (the nearby city, which could rival Gotham in terms of average crime). He rebrands as Nightwing, named after a myth he heard from Superman. He and Batman aren't on good terms during this time, especially because Batman gave some random kid his old identity. Though he is rightfully angry with Bruce, his relationship with Jason is okay. They're not the closest, but they do seem to get along reasonably well (they went on a ski trip). Jason dies while Dick is on a mission in space, and he finds out too late to come to the funeral. Dick actually hallucinates Jason a couple times after his death. When Tim asks him to be Robin, he refuses, but helps train Tim after.
Things in Bludhaven aren't great. He joins the police to try and weed out its corruption, but isn't able to actually do too much. The city is bombed by Deathstroke, though they have rebuilt. He also fought a villain named Blockbuster, who knew his idenity. Blockbuster blows up his apartment, and kills anyone Dick talks to. Blockbuster is eventually killed by Tarantula, someone Nightwing was working with, and he tries to turn himself in from guilt. Dick doesn't spend all his time in Bludhaven anymore, and splits time between there and Gotham. One good thing during his time there was that he found and adopted a three legged dog, who he named Haley (after the circus he grew up in).
At some point, Bruce is presumed dead and Dick takes over as Batman. Damian has recently arrived, so Dick is in charge of taking care of him. After Alfred gives Damian the Robin suit, Dick decides to keep him as Robin (which has the benefit of keeping him here, and not with assassins). Tim believes Bruce is still alive, but Dick just thinks he should get therapy. They fight, and Tim leaves. Dick and Damian protect the city together, and Dick teaches Damian to be more than just an assassin. After Tim finds Bruce and brings him back, Dick saves him from falling and returns to being Nightwing.
One last part of his story is the Spyral timeline. Bruce doesn't act very nice, and it could be ignored as being out of character and weird, but it is a major part of Dick's life and experience. It starts with the Crime Syndicate capturing Dick, and hooking him up to a bomb that runs off his heart. He's killed by Lex Luthor to stop the bomb, and his heart is restarted after Batman freaks out. While he's recovering, Batman tells everyone he died, and they hold his funeral. When Dick wakes up, Batman tells him he needs to go undercover, because a new organization (Spyral) knows their identities. Dick disagrees with not telling his family he's alive, and he and Bruce fight. Dick ends up going on the mission, and has a cool (but also horrible) time as a spy, and he's sent on missions to collect organs from people with superpowers. Batman has memory loss, and Dick loses his only contact with the outside world. Eventually, he suceeds in bringing them down and is able to return to Gotham. Tim and Jason are mad at him for faking his death, and he never tells them the truth. I think they're getting along again though.