By now, it's widely accepted knowledge that the third Summers brother is Gabriel Summers, the X-Men's temperamental powerhouse Vulcan. However, this wasn't always the case, and the identity of the third Summers brother -- or if he even existed -- was one of the biggest X-Men mysteries of the 1990s.
The plot was first revealed in 1993's X-Men #23, by Fabian Nicieza and Andy Kubert. In Alaska, Mister Sinister (who has a long obsession with the Summers bloodline) has a conversation with Cyclops. Sinister offhandedly mentions that Scott has "brothers" although, he backtracks once Scott questions that. At the time, Alex was believed to be Scott's only sibling.
Hints of the original plot points can be found throughout Nicieza's work on the franchise, including in X-Men #39. This story features Adam X meeting Philip Summers (Scott and Alex's paternal grandfather) in Alaska. Despite Adam's lineage not being related to the paternal side of the other Summers brothers, for some reason, he feels a familiarity with Philip. However, this was ultimately just a conincidence.
For a while, Gambit was a character who was teased and considered to be an official Summers brother. At the time, his origins before being adopted by the Thieves Guild were vague. When Sinister made that cryptic remark in X-Men #23, Gambit was a largely viable candidate.
In Chris Claremont and Sean Chen's X-Men: The End, that idea became a reality in the alternate future timeline of that 18-issue maxi-series. In this series, Gambit was a mutant created by Mister Sinister. Sinister spliced his DNA with Cyclops' in an effort to create a mutant that could defeat Apocalypse. When Apocalypse learned of Sinister's plan, he ensured that the child was stolen and left in the hands of the Thieves Guild, and the child then grew into Gambit.
While Adam X was initially supposed to be the son of D'Ken and Katherine, that idea evaporated when Ed Brubaker and Trevor Hairsine introduced Vulcan as the third Summers brother in 2006's X-Men: Deadly Genesis.
Before Christopher Summers became the Starjammers' Corsair, he and a pregnant Katherine Summers were captured by the Shi'ar. Gabriel Summers was that baby, effectively retconning Adam X's possible origin out of existence. If Katherine was already pregnant with Gabriel and died before Adam was born, there was no way she could be Adam's mother.
Despite Gabriel's parentage, he wasn't raised by the Summers. After discovering her pregnancy, the Shi'ar removed the fetus from Katherine's body and placed it into an incubation accelerator. This allowed Gabriel to age at a rapid rate and soon, the Shi'ar sent him to Earth. Initially, he was supposed to become a slave for Erik the Red, but Moira MacTaggart found him and took him in instead.
Charles Xavier also took notice of the child, realizing that he shared many similarities with Scott. Alongside Petra, Darwin, and Sway, Vulcan became an X-Man and was tasked with the mission to save the X-Men from Krakoa. The mission didn't go as planned, and Vulcan ended up as one of the sole survivors, developing a deep hatred for the Summers family after he was left in space.
Xavier also wiped all memories of Gabriel's existence from Scott's mind, leaving him effectively nonexistent to the X-Men. For a while, Gabriel was a major villain on the cosmic side of the Marvel Universe, ruling the Shi'ar Empire and causing disarray in the lives of his family. Now, he seems to be on better terms with his family and lives with them in the Summer House in Johnathan Hickman's X-Men.