Classic Mode is the main mode of the Super Smash Bros. series in each of its installments. The mode is usually different for each fighter.
In Super Smash Bros., the mode is named 1P Game. Metal Mario is a unique fighter in this mode, as Metal Boxes would only debut in the second game, and is even given his own stage to fight on. The stages are always accessed in a consistent order, with the Mario franchise-related battles being the Yoshi Team (stage 2), Mario Bros. (stage 4), Giant Donkey Kong (stage 6), and Metal Mario (stage 9). Four of the battles take place against Mario fighters, with one of them specifically featuring Metal Mario and a unique stage for him (Meta Crystal). Classic features a different congratulations message for each fighter at the end.
In Super Smash Bros. Melee, the mode is named Classic, and while the characters and stages are now randomized, the set up remains the same (like how Master Hand is fought at the end or that Snag the Trophies is the sixth stage). The opponent fighters are always fought on a specific stage representing their franchise (e.g. Mario is fought at either Princess Peach's Castle or on Rainbow Cruise & Luigi is always fought in one of the Mushroom Kingdom stages). A majority of the Giant battles have more than 50% chance of fighting a Mario character (Mario, Yoshi, Dr. Mario, Luigi, Donkey Kong, and Bowser), and they make up 6 of the 10 possible opponents (the others being Jigglypuff, Captain Falcon, Link, and Young Link). There is a Metal battle, and the scripted fight with Metal Mario has been moved into Adventure mode. By completing Classic with a fighter, the player can earn a trophy of that fighter.
In Super Smash Bros. Brawl, Classic Mode's layout is randomized per franchise (e.g. a Yoshi or a DK battle always occurs on stage 2). Just like in Super Smash Bros. Melee, there are a few Mario-related unlock rewards (aside from the character-specific trophies), such as the Paper Mario trophy or Luigi in an unlock match if no continues have been used. The Yoshi, Wario, and Donkey Kong franchises are also part of the randomization.
In Super Smash Bros. for Nintendo 3DS, Classic Mode has paths, many of them being marked by a random franchise symbol, giving more control as to which path to take. For instance, if one were to take the path with the DK symbol, the opponent and stage from that franchise is selected. In Super Smash Bros. for Wii U, Classic is set up like a tournament, and while the game randomly selects the fighters, one can see whom they can fight before starting the battle (and a faint preview of the stage where the battle occurs is shown beneath them). The opponent and stage are not always from the same series, even in single-opponent battles (e.g. there's a possibility of fighting Mario on a The Legend of Zelda stage).
In Super Smash Bros. Ultimate, every fighter has their own predetermined route, each having a distinct title (e.g. Donkey Kong's Classic Mode route is titled "Journey to New Donk City"). A majority of the fighters fight Master Hand at the end. In the case of Mario and Captain Falcon, the final boss is Giga Bowser.