The original Flash was Jay Garrick, created in The Golden Age of Comic Books. He fell into obscurity and the Super-Speed superhero concept was revamped into a new character, Barry Allen, in The Silver Age of Comic Books. Jay existed as a comic book character in this version, and Barry took the name from him. Later, Barry met a Jay who existed for real on another Earth in a story that established The Multiverse of DC. Barry's nephew Wally West also became Barry's similarly powered sidekick Kid Flash. After Barry died in the Crisis on Infinite Earths reboot, Wally took up his mantle. The Cosmic Retcon event also established Jay was a real person who still had inspired Barry. Wally served as the Flash for many years until the mantle passed to Barry's grandson Bart Allen. Then it passed to Wally again after Bart was killed. Then Barry came back from the dead and he and Wally shared it. Then the universe was rebooted again, and Barry became the only Flash on the mainstream Earth. And with DC Rebirth, the original Wally has returned from the Speed Force as the Flash, once again sharing the name with Barry. And a while after that, Avery Ho would become China's Flash, although she was initially only tangentially associated with the main Flash family. Jay Garrick would eventually make an appearance as well, having be retconned out of existence. With the events of Speed Metal, Wally West has officially returned to the role of The Flash since the end of Final Crisis.
In The New 52: Futures End, the New 52 version of Wally West (later retconned into being the cousin of the original Wally) became the second Flash in that timeline.
The Kid Flash identity. The original is Wally West, who retired from superheroics after a fashion but briefly returned to it during Crisis on Infinite Earths. That same story saw him take up the Flash identity. Decades down the line, his cousin and Barry Allen's grandson Bart Allen, then going by Impulse, took up the Kid Flash identity. Then the New 52 happened, and Bart (or Bar Torr) was established as the first Kid Flash, having never been Impulse. He wasn't well-received and went off into Comic Book Limbo, and by that time, a new version of Wally West, named Wallace R. West, was introduced. In DC Rebirth, Wallace West officially became the new Kid Flash...the third one, because by that point, the original Wally West returned, and was established as having been the first Kid Flash.
In Kingdom Come, Iris West II, the daughter of Wally, is the second Kid Flash, having taken up her father's mantle.
Impulse was the name used by Bart Allen as a way of side-stepping being the new Kid Flash. He eventually did become Kid Flash, and years later, Wally West's daughter, Iris "Irey" West II gained super speed and picked up the Impulse codename. Both Bart and Irey would be removed for existence due to the New 52. With the return of Bart Allen in Young Justice (2019) whilst Irey continued to exist in limbo, he regained the mantle of Impulse. As for Irey, she would casually continue to use the Impulse mantle alongside Bart until the events of One-Minute War where she would christen herself as "Thunderheart"
Much of the Wally West's Rogues Gallery consisted of successors to the original — in one case a son, but usually just another criminal who somehow got hold of the original's gear. The second Mirror Master, Evan McCulloch, an assassin who was eventually hired by the FBI and given the first Mirror Master Sam Scudder's technology so he could take out high-value targets while pretending to be a supervillain. He eventually got sick of the work, killed his superiors, and went freelance, eventually winding up in the Rogues and earned their acceptance.
Similarly, the the second Trickster, Axel Walker, is a teenager who stole the technology of Jesse James, the first Trickster. James had retired by this point, becoming an FBI agent, and was actively planning to shut down the Rogues. He despised Axel and after some mind control stuff, took back the mantle of the Trickster. However, after the Rogues got Bart Allen killed, James would end up in some trouble and eventually was killed. Axel still used the Trickster identity in this time, and the Rogues let him tag along, and grew to like him.
This was taken to the point of parody with Chillblaine, an identity taken by a succession of dumb but good-looking crooks with Captain Cold's gear, all "recruited" by Cold's sister, Golden Glider. Eventually she made the mistake of choosing someone who was actually dangerous and he killed her. Cold came out of retirement to avenge her, and ended up back in the role of the Rogues' leader.
The 25th century features the Renegades, a police task force that utlizes the weaponry of the present day Rogues.
Captain Cold -> Commander Cold
Heat Wave -> Heatstroke
Mirror Master -> Mirror Monarch
Weather Wizard -> Weather Warlock
Trickster II -> Trixster
Golden Glider -> Golden Guardian (also a member of the Sinestro Corps)
This was actually inverted in one case, with the Reverse-Flash. The first (second if you count Jay Garrick's own Evil Counterpart, The Rival) Reverse-Flash to be shown in the comics is Eobard Thawne, a time-traveling criminal from the 25th century. However, the next Reverse-Flash to be shown, Hunter Zolomon, was born in the present day and took on the identity of Zoom after being inspired by a discussion he previously had with Wally West about Thawne. When Thawne himself eventually shows up again via his usual Time Travel antics to team-up with Zolomon, he's quite amused.
In the New 52, a fourth Reverse Flash would debut, this time being Wallace West's father and Iris's younger brother, Daniel West. Daniel would later die being involved with the Suicide Squad. Eobard Thawne would continue exist in this timeline, this time going purely with his original title of "Professor Zoom". With the events of DC Rebirth, the Pre-Flashpoint Eobard Thawne overwrote the New 52 version, returning to the title of the Reverse Flash again.
Ironically enough, the Flashes and Reverse Flashes share a biological legacy. Barry Allen has an older twin brother named Malcolm Thawne. Malcolm's adoptive mother's child was stillborn. During the birthing process of Nora Allen's twins, a blackout occurred. In order to cover up the death, the doctor who was delivering the babies decided to give one of Nora's twins to Charlene Thawne, switching him out with the stillborn one and lying to Nora that one of their children that they had died. Malcolm would grow up harnessing the power of the Blue Flame, given to him by his adoptive grandmother to become Cobalt Blue. At one point in time, Malcolm found out the truth from his parents and told Barry. Jealous of his employment as a policeman, Malcolm got employed as a janitor at the CCPD to spy on him. Witnessing the accident that gave Barry superspeed, Malcolm would grow to hate Barry and become a supervillain. He would then plague Barry and Kid Flash for years to come. His descendants would come to hate the Allens as well, creating an eternal conflict between the two families. His most famous descendent being Eobard Thawne, the Reverse Flash. The two dynasties would merge together when Barry's son, the Tornado Twin Don Allen, and Eobard's descendent Meloni Thawne fell in love, eventually having a son, Bart Allen aka Impulse.
