One Piece is at a crucial juncture in its story right now, with Luffy and his allies facing down two of the Yonko (or Four Emperors), the most powerful pirates in the world. Chapter 1040 saw the momentous defeat of one of these emperors, Charlotte Linlin, better known by her epithet "Big Mom".
Big Mom's defeat by Luffy's allies (for the time being) Trafalgar Law and Eustass Kid, has major implications for the future of One Piece. Currently, it's unknown if she is alive or dead, and the answer will affect future events significantly.
Who is Big Mom & What Happened to Her?
Big Mom is the only female Yonko, and is consequently one of the four most dangerous pirates in the world. Her bounty of 4,388,000,000 berry is the fourth-highest ever confirmed, and if alive the second-highest for a living pirate after her ally Kaido (who trails only the deceased Whitebeard and Gol D. Roger). This massive bounty is due to a few factors. On top of being a pirate, Big Mom is the queen of Totto Land, an archipelago in the New World, and has attempted to maintain control over further territories like Fish-Man Island, making her a threat to the World Government. Her nation and subsequently her crew is made up primarily of her 85 children.
Aside from her family, Big Mom's military force stems from a few non-related allies and, importantly, the Homies, servants she creates from inanimate objects with her Devil Fruit, the Soru Soru no Mi or Soul-Soul Fruit using others' life force or even her own. Big Mom controls three special Homies that function as her primary weapons in battle: Napoleon, her sentient hat, can turn into a giant blade, Prometheus is a living ball of fire, and Zeus and later Hera are living storm clouds which generate lightning. On top of her Devil Fruit, Big Mom is a powerful user of both Armament and Conqueror's Haki, and is physically one of the strongest characters in One Piece.
Big Mom eventually made an alliance with Kaido in Wano to search for the Pirate King Gol D. Roger's treasure, the eponymous One Piece. This pitted her against Luffy and his allies for a second time, after they initially clashed when Luffy interfered in the wedding of Sanji to her daughter Pudding (which turned out to be an assassination plot directed at Sanji's family). While Luffy fought Kaido, eventually Big Mom fought and Eustass Kid, who like Luffy and Zoro are classified with other powerful young pirates as the 11 Supernovas.
The battle exhausted both sides, forcing Big Mom to create a new Homie called Misery from Prometheus and Hera's powers, and eventually opened a hole in Kaido's floating fortress of Onigashima as well as the ground below. Kid and Law combined their powers in a last spurt, specifically using Kid using a giant railgun to propel her downward while Law sliced Misery in half with his Ope Ope no Mi to prevent her from interfering. She fell through the hole, and accidentally set off a cache of explosives as she fell, which followed her as she crashed to the ground, defeating her.
Is Big Mom Alive?
Big Mom's last thoughts before her defeat involved her lamenting Gol D. Roger not divulging more about the One Piece to her, as well as indirectly sending many driven young pirates after her. She also vowed that she would not die, and would get revenge on Law and Kid. In all likelihood, Big Mom is not dead. Big Mom's durability and sheer power also make it unlikely that she would die even from the large explosion that defeated her, especially because she didn't fall into the ocean, which is far more dangerous for Devil Fruit users.
Series precedent also strongly points to Big Mom surviving her defeat. No major villain in One Piece has lost their life in battle over the series' quarter-century run. Most major villains defeated in the series are either arrested by the Marines or move on with their lives, their various plans foiled. Some remain semi-major players in the story despite their defeat, like Rob Lucci and Buggy the Clown.
If Big Mom is alive, she wouldn't be arrested by the Marines because Wano, the site of her defeat, lies outside the World Government's jurisdiction. That said, it is likely that if Kaido also loses that they will be forced to leave Wano, and possibly move their base of operations back to Totto Land before continuing their quest for the One Piece. This would stop Kaido's ambitions of controlling Wano and place it back in the hands of the Kozuki family.
What if Big Mom is Dead?
Though it's the less likely scenario, Big Mom's death would have immense repercussions in the world of One Piece, similar to how things changed drastically after Whitebeard's death at Marineford. In the latter's case, his crew fragmented entirely after his death, with most members' whereabouts unknown. This is less likely to be the case for Big Mom, as her crew is mainly composed of her children, and they retain a base of operations.
That said, there may be infighting about who should succeed her. Major contenders are her eldest son and child Perospero, and the Three Sweet Generals Cracker, Smoothie, and Katakuri, who are her strongest children. Perospero in particular disapproved with his mother's alliance with Kaido, and may choose to end it if he takes over her crew (although doing so could risk war with Kaido's slightly stronger crew). It's also unknown what would happen to the thousands of Homies created by Big Mom's Devil Fruit should she die.
Outside what it would do to her family and domain, Big Mom's death could also indirectly strengthen the World Government and diminish Kaido's political power, while perhaps bolstering the power of the other two Yonko, Blackbeard and Shanks. Just as significantly, the Soru Soru no Mi would reappear as a Devil Fruit, and someone else could eventually gain its power.
Regardless of Big Mom's fate, Kid and Law have now cemented themselves as Yonko-level pirates, and their bounties and influence will probably increase dramatically. If Kaido is defeated, it also spells a formal end to the alliance between the two and Luffy and may mean competition between all three captains in the chapters and episodes ahead. All of this points to the fact that Big Mom's defeat has changed One Piece completely.