The Long Version (thank you, Wikipedia):
Over a number of decades in the 20th century, DC Comics reimagined its staple characters and concepts and acquired comic book properties from rival publishers. Classic characters which had been replaced were then brought back, and explained as residing on Earth-Two, while those of other publishers lived on fanciful worlds such as Earth-S and Earth-X. The Charlton characters (including Captain Atom) resided on Earth-4.
This storytelling device, positing an infinite “Multiverse” of worlds full of heroes and villains, fed many DC stories for some time. In 1985 however, DC chose to simplify its continuity with a crossover story called Crisis on Infinite Earths, in which all of the Multiverse was combined into a single world with one amalgamated history. Subsequent “crisis” storylines have seen reality threatened and reshaped again, often as a way of adjusting the fictional history of the setting of DC’s stories.
Stories like this include Zero Hour: Crisis in Time, Infinite Crisis (which brought back the Multiverse concept, this time with 52 finite worlds). Final Crisis, and Flashpoint (which resulted in the 52-world Multiverse resetting, and bringing about the contemporary setting of DC Comics, published until Convergence under the branding “The New 52”). The Grant Morrison comic book The Multiversity also explored the complete meta-history of the Multiverse in further depth, and mapped out a majority of the 52 worlds which make it up since Flashpoint.
In the run-up to Convergence, DC published two weekly series, Earth 2: Worlds End and The New 52: Futures End, which led directly into the crossover story. Additionally, the 2014 Futures End Booster Gold one-off comic depicts the classic Superman villain Brainiac, having assimilated his parallel counterparts from across the Multiverse, extracting from time travelling hero Booster Gold the location of Vanishing Point, a moment at the end of the universe from which all of history, including prior to each of DC’s crises, can be accessed. From there, he sets about collecting various cities on various worlds prior to those worlds’ destruction; he attempts to do this to a possible future of DC’s main New 52 setting in the narrative of Futures End, but is narrowly defeated by that world’s combined heroes and imprisoned.
Spinning out of the events of Earth 2: Worlds End, the heroes of Earth 2—Batman (Thomas Wayne), Superman (Val-Zod), Green Lantern (Alan Scott), and the Flash (Jay Garrick)—find themselves thrust onto the desert landscape of the planet Telos, along with Yolanda Montez (a former host for the power of the Red, the collective animal life force of Earth 2) and reporter Dick Grayson. There they encounter what appears to be the villain Brainiac, whose shape changes from panel to panel to reflect different incarnations of Brainiac from over the years; it is later revealed that this character is Telos, an envoy of Brainiac who is the living embodiment of the planet, who awaits his master’s return.
Just as their world was about to be destroyed by Darkseid, the Earth 2 heroes (known as the Wonders of the World) were miraculously transported to Telos, where Brainiac, who combined with and assimilated his incarnations from across the Multiverse, has captured and stored many cities from across the entire history of the Multiverse, spanning various eras of DC Comics’ publishing history, along with the settings of many of its popular Elseworlds graphic novels. After a brief skirmish with the heroes, and wondering what purpose his master Brainiac envisioned for this world of forgotten cities, Telos enacts a plan to make all the captive heroes and villains of the Multiverse fight each other for supremacy.
These battles are shown in various two-part issues during the two-month Convergence series of comic books.
The Earth 2 heroes investigate their setting, and Yolanda Montez discovers she has residual animal-like abilities, such as sharp claws and reflexes, from her time as an avatar. While investigating the planet, Batman partners with Dick and enters a version of Gotham City, surmising he will know his way to the Batcave. He enters the pre-Flashpoint Gotham on the former Earth 0 (the primary setting of DC stories prior to its New 52 relaunch) where Thomas Wayne (who became Batman following the death of his son) the and pre-Flashpoint Batman, Bruce Wayne, share an emotional encounter with one another. Bruce lends Thomas his Batmobile, but they are pursued out of the city by some of Gotham’s villains. Thomas Wayne kills himself with an explosive device, taking the villains with him. Recognising from what he has seen on Telos that on many worlds he is either a Batman himself or closely connected with Batman, the mournful Dick Grayson picks up one of Thomas’s Batarangs and decides to follow in his footsteps.
Alan Scott uses his powers as an Avatar of the Earth to learn that deep within the planet Telos is the preserved world of Skartaris (the setting of classic The Warlord comic books) where they encounter Warlord’s nemesis Deimos, who promises the heroes he can defeat Telos. The Wonders soon learn however that Deimos has systematically captured the powerful time traveller group, the Time Masters, as well as extremely powerful time travelers such as Monarch and Per Degaton. He absorbs their collective power and uses it to summon Brainiac (who was imprisoned in a T-sphere in The New 52: Futures End).
Deimos banishes Brainiac, and then reveals to Telos that he is not the planet incarnate, but a mortal who was transformed into a Brainiac-like creature when Brainiac ravaged his homeworld to create a storage world for his captured cities. Telos, shocked, vanishes, and Deimos declares himself the planet’s ruler. Using his near-infinite power, he tells the captives of the planet that he will protect their lives for a price.
As this is happening, planet Telos manifests physically in DC’s contemporary main setting of Earth 0, where Superman, Supergirl, Guy Gardner and the Red Lanterns, the space-faring Justice League United and the main Justice League all watch on with amazement. Even all-powerful beings such as Nix Uotan, the guardian of the Multiverse, the evil god Darkseid and the cosmic being known as Oracle do not fully fathom the events they are witnessing.
Meanwhile, on Telos, the Earth 2 heroes encounter several pre-Flashpoint heroes (Superman, Red Arrow, Batgirl (Stephanie Brown), Donna Troy and the Flash (Wally West)), with whom they form a stronger team. Superman tells Dick Grayson he heard his interactions with Telos, and knowing the Dick of his world, places faith in him that he can recruit Telos to their side to challenge Deimos. The two Flashes, Jay and Wally, begin to search Telos for other Flashes, and add the second Flash (Barry Allen) from the Silver Age of Comic Books to their team. They are soon joined by versions of heroes and villains from comics published in every decade of DC’s history and plan to overpower Deimos.
However, Deimos reveals he has recruited various villainous and misguided DC characters to defend him in exchange for giving them their deepest desires, including characters from the Flashpoint apocalyptic timeline, the Kingdom Come graphic novel, the Marvel Comics-inspired Extremists, and the classic Golden Age Crime Syndicate of Earth-Three.
However, when Yolanda reveals to the villains that they are being used by Deimos who wants them to die to grant him “the magic of death” to remake the Multiverse in his own image (knowing this has happened several times in DC’s publishing history), the two teams unite. Telos, convinced by Dick to fight for the heroes, battles it out with Deimos and weakens him. Deimos is then finally destroyed by the Parallax version of Hal Jordan from the Zero Hour storyline. This inadvertently releases all the energy of the Time Masters into the world, creating an arc of lightning which escapes Telos into the space of DC’s main Earth 0 setting striking the omniscient Oracle character, who breaks up. As he dies, Oracle explains that the whole of reality is coming apart, as in previous DC crises.
Seeing no other solution, the time traveler Waverider summons Brainiac back. Brainiac reveals he was the pre-Flashpoint Brainiac who surviving that event sought to explore the history of the Multiverse, but became caught in that history and mutated by the effects of the previous crises. He realizes what a monster he has become and wishes to put things right. He prepares to send everyone back where they came from, but is prevented by the damaging effects of the original crisis in Crisis on Infinite Earths. The heroes resolve to change the outcome of that Crisis. As Brainiac sends the pre-Crisis Supergirl and Barry Allen back to meet their fates in the original Crisis story, pre-Flashpoint Superman and Zero Hour Parallax (seeking redemption) volunteer to go with them, changing the outcome of the crisis. Brainiac explains that this has brought changes to the Multiverse, causing everything to “return to what it was before I [Brainiac] brought you here.”
The old worlds of the classic Multiverse live on in both their original versions and the modernized forms depicted in The Multiversity. All worlds now exist. The Earth 2 team are left behind on the planet, but Telos transports the planet into Earth 2’s universe. He appears in the sky to tell the Wonders that he has remembered his real name and where his family is and that the planet will be their New Earth 2. Once again able to channel the power of the Green, Alan terraforms the planet and makes contact with the space fleet carrying the displaced Earth 2 refugees leading them to their new home.
The aftermath of Convergence saw an end to The New 52 branding, however the continuity established by The New 52 did continue.
Captain Atom appeared throughout the Convergence series (primarily in the background) in both his Modern Age and Bronze Age incarnations. He also appeared in Convergence: Justice League International #1 & #2 in his Modern Age form, and in Convergence: Blue Beetle #1 & #2 in his Bronze Age (Earth-4) form.