A home stage is a stage or set of stages associated with a character by certain game modes.
Characters' home stages are normally from the same universe as the character. However, in all the Super Smash Bros. games there are playable characters who do not have a stage belonging to their universe; such characters are instead assigned a home stage from another universe.
Super Smash Bros.
Each of the eight starter characters has their own home stage, while the four unlockable characters share stages with starter characters. All starter characters and two of the unlockable characters (Luigi and Jigglypuff) have home stages from their own universes, because Jigglypuff and Luigi are from the Mario and Pokémon universes, they have (Peach's Castle and Saffron City) as their home stages (sharing it with Mario and Pikachu respectively). Because there are no stages from the EarthBound and F-Zero universes, the other two unlockable characters (Ness and Captain Falcon) have home stages (Dream Land and Planet Zebes) from different universes, as those two stages take place when unlocking these two unlockable characters.
Mushroom Kingdom, an unlockable stage from the Mario universe, is not the home stage of any character. However, it is sometimes considered the home stage for Mario and Luigi (due to the universe it is from). But it is never fought on in 1P Game. Even if said stage is unlocked, Stage 4 still takes place on Peach's Castle.
Super Smash Bros. Melee
In Super Smash Bros. Melee, characters have different home stages in different game modes. All characters have a home stage for All-Star Mode, most have one or two home stages for Classic Mode, and unlockable characters have a stage for their unlock challenge fight.
All universes with playable characters have at least one Melee-introduced stage (i.e. non-past stage), except Fire Emblem. However, even for characters that have a stage from the same universe, in some game modes they will use a stage from a different universe as their home stage.
Super Smash Bros. Brawl
All characters use all of the Brawl stages (i.e. non-Melee Stages) from their own universe as their home stages, with the exception of R.O.B. and Mario Bros. There are no stages from R.O.B.'s universe, so instead R.O.B. uses Mario Bros. as his home stage (while Mario universe characters do not use it as their home stage).
Super Smash Bros. 4
Unlike in previous games, Past Stages also serve as home stages. All characters' home stages are all of the stages from their own universe, unless there are no stages from their own universe. Characters from universes without stages have all of the stages whose universes lack characters as their home stages. The Mii Fighters' home stage is considered to be Battlefield, as they are part of the Super Smash Bros. universe and are only fought in Classic Mode as enemy teams.
In Super Smash Bros. for Wii U, Classic Mode pits fighters on random stages, with no correlation in most cases. However, Home Stages remain as a concept in All-Star Mode.
Super Smash Bros. Ultimate
In Super Smash Bros. Ultimate, home stages are absent as a concept ingame, due to Classic Mode setting a single stage for each fight instead of randomly by character, and All Star Mode allowing the player to choose the stage. However, the Super Smash Bros. Ultimate Official Site still acknowledges the concept by having a stage in the background of each fighter's page. In many cases, these stages are where that fighter is unlocked, while for downloadable characters, it is the stage they are bundled with.
Hypothesis of "home advantage"
There is no clear or consistent benefit to a character fighting on their "home stage." Any advantages that certain characters have over others on stages happen across the entire stage list and are situational; they are not affected by the universes in which the characters originate from (for example, Fox and Pikachu dominate on Temple, even though they do not originate from the Zelda universe). In some cases, it may be to a character's disadvantage to be fighting on their home stage; for example, Ness is unable to use his recovery move when knocked between buildings in Fourside. Additionally, Final Destination, which acts as the home stage for Roy, is disadvantageous for him against nearly every character (except for arguably Fox and Falco) as he is much more vulnerable to combos without any platforms to escape to. In Brawl, the ledges of the Lylat Cruise stage are thinner than most other stages, causing Fox, Falco, and most notably Wolf to curve underneath the stage more frequently when using their respective recovery moves instead of directly grabbing the edge. Additionally, in Skyworld, Pit's Centurions in his Final Smash will crash into the breakable platforms when they aim for opponents, quickly destroying the stage and temporarily removing all ledges; this consequently removes Pit's ability to plank and makes him much easier to gimp.
However, in contrast, some stages do offer advantages to some characters; as mentioned before, any such advantages are non-circumstantial, but these certain stages may have been the origination for the "home advantage" hypothesis. In Melee, Fountain of Dreams serves as Marth's home stage, and is frequently cited as one of his best stages in the entire game, even at high-level tournament play. The stage's small size allows him to control space more efficiently, and the extended bottom blast line allows him more room to recover with Dolphin Slash, which travels purely vertically. In Brawl, on Bridge of Eldin, Zelda and Sheik's Light Arrow, Ganondorf's Beast Ganon, and Toon Link and Link's Triforce Slash are much easier to use, and harder to dodge; the frequent absence of a lower blast-line also acts as a benefit to Link, Ganondorf, and Sheik, who normally have relatively poor recoveries. The low blastline of Shadow Moses Island acts as a benefit to Snake, as most of his attacks (especially his special moves) have mostly upward knockback; his high weight and falling speed make him more resistant to vertical KOs, so he is not as affected.
On other stages, characters can receive neither a clear advantage nor disadvantage. On Summit, for instance, all the characters slide around, except for the Ice Climbers, who experience no change in their friction when walking on the stage. Like other characters, however, they are still unable to grab onto the ledges in the stage; this trait causes Belay to more frequently cause Nana to be KO'd, thanks to Belay giving Popo more height than Nana when there is no ledge to tether to. Another example is Little Mac on Boxing Ring: the stage gives no access to the bottom blast line, which remedies Mac's poor recovery, but also has plenty of verticality, making it difficult for him to traverse the stage. In Super Smash Bros. 4, the Ω form of Boxing Ring takes this even further in Mac's favor — while being based off of Final Destination means it has a lower blast line, the walls of the actual stage reach all the way down to it, allowing Mac to make full use of his uncharacteristically superb wall jump.