Home stage
A home stage is the stage that is most commonly associated with a character, by the game and/or by players. The subject of home stages can at times be somewhat confusing in the Super Smash Bros. series. In all the Super Smash Bros. games, there are playable characters who do not have a stage belonging to their "universe". In addition, certain characters in SSBM have more than one stage. As secret characters are fought on stages specific to each of those characters, and SSBM's All-Star mode also gives each character a "home" stage, it's possible to determine which characters "own" which stages, even though the results may occasionally cause confusion.
Super Smash Bros.
In Super Smash Bros., each of the eight starter characters has their own stage. Of the four secret characters, two - (Luigi and Jigglypuff) - share their stages with the starter characters from their universe (Mario and Pikachu, sharing Peach's Castle and Saffron City, respectively). The remaining two characters have their home stages defined by where they are fought when they are unlocked. Captain Falcon is fought at Samus's Planet Zebes, while Ness is unlocked by defeating him at Kirby's Dream Land stage. Thus, Captain Falcon and Ness may be considered to have these stages as their home stages. There is another stage from the Mario series, Mushroom Kingdom, which is sometimes considered as Luigi's stage, as well as Mario's.
Super Smash Bros. Melee
In Super Smash Bros. Melee, the situation is somewhat more complicated. While secret characters all have a stage of their own for the purpose of unlocking them, all characters have a stage associated with them for the purposes of All-Star Mode. However, for series origin purposes, Marth and Roy have no home stages, the only characters not to have one.
Super Smash Bros. Brawl
In Super Smash Bros. Brawl, stages and characters are not matched one-to-one like they are in Melee. What follows is a list of home stages based on where characters are fought in Classic Mode and All-Star Mode, almost always based on a character's universe. The only exception is ROB, who has no stage from his universe. Because of this, he has no true home stage, though as he is fought at Mario Bros in Classic and All Star modes, while Mario characters are never fought there in those modes, that stage can be considered his.
As was the case with the Past Stages in Melee, stages returning from that game are never seen in Classic or All-Star Mode. This leads to the fact that Halberd is always seen in Classic Mode, no matter what, since that is the Kirby series' only Brawl stage, and the Kirby series is always fought in Classic Mode.
Home advantage
There is no clear or consistent benefit to a character fighting on his "home stage." In some cases it may be to a character's disadvantage to be fighting on his 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 frequently considered his primary counterpick, as he becomes considerably easier to combo, especially against fast-fallers. In Brawl, Donkey Kong's Final Smash makes him immobile; if used on Rumble Falls, a scrolling stage, it will possibly result in a self-destruct, with the exception if he is airborne while doing the attack.
However, in contrast, some stages do offer advantages to some characters. For example, on the Bridge of Eldin, Zelda's Light Arrow 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, who normally has a poor recovery. Similarly, on Summit, all the other characters slide around, except the Ice Climbers, who are the same as if they were on any other field, although, like other characters, they are still unable to grab onto the ledge.
