Hitstun is a period of time after being hit by an attack that a character is unable to act outside of directional influence or teching. It is directly dependent on knockback, so at a same damage percentage, an attack like Falcon Punch will have more hitstun than one like Thunder Jolt. When a hit deals insufficient knockback to lift the opponent off the ground, hitstun is equal to the length of the flinch animation, while at higher knockbacks it is a straight multiplier of the knockback received. Each Smash game has a programmed value that is multiplied by the amount of knockback received to determine the amount of frames a character is locked in hitstun after being hit; for example, Melee has a hitstun multiplier of 0.4 frames per unit of knockback, so a hit that deals 100 units of knockback will leave the target in hitstun for 40 frames.
Hitstun is an essential component of combos, as the basis of a combo is to have enemies trapped in hitstun while constantly being attacked. As a result, games that have more hitstun for equivalent knockback open up more potential combos.
The amount of hitstun in the original SSB is much higher than in the subsequent Super Smash Bros. games, being a x0.533 multiplier. The multiplier is so high that long, highly damaging combos are commonplace, with competitive matches being heavily centralised around them. Zero-death combos are not uncommon in SSB, as every character in the game except Samus is capable of pulling a zero-death using their normal moves. The commonness of zero-deaths in this game is one of the primary reasons why competitive matches are played with 5 stock, more than Melee, despite the game not being as fast-paced as Melee.
The amount of hitstun in Melee is a significantly lower x0.4 multiplier, making Melee a less combo-oriented game, and zero-death combos a rarity. However, with the increased falling speeds, and generally faster and more varied movement, combos remain frequent, though generally more difficult to pull off consistently.
Brawl has the same hitstun multiplier Melee has. However, when hit, characters can now air dodge after 13 frames and use an aerial attack after 25 frames out of hitstun, regardless of the actual amount of hitstun they sustained. This new mechanic, known as "hitstun canceling", drastically reduces the amount of actual hitstun characters have to sustain upon being hit, especially at higher knockback. For example, while a character sustaining 100 units of knockback in Melee will be in hitstun for 40 frames, that character will be in actual hitstun for only 13 frames in Brawl if they air dodge or 25 frames if they attack, and if that knockback value is doubled, the Melee character would sustain an inescapable 80 frames of hitstun, while the Brawl character will still only sustain 13 or 25 frames of hitstun.
This change drastically reduces the amount of true combos, and means the only attacks that can truly combo are those that deal very low knockback while having low enough ending/landing lag to followup within 13 frames before the opponent can air dodge (such as chain throws, Meta Knight's up aerial, and Sheik's forward tilt). Any attack that deals even slightly more knockback or has slightly more ending lag will require the player to read the opponent's action to successfully followup. This new mechanic also allows launched characters to act before their knockback wears off, which allows characters to perform certain actions during knockback to reduce their aerial momentum, allowing them to survive blows that would have otherwise KO'd.
The much slower falling speeds, loss of L-canceling with most aerials not having their landing lag compensated, and generally slower movement also reduces the amount of possible combos, though the effect of these is minuscule compared to the ability to act so early out of hitstun.
This new mechanic is one of the most controversial additions to Brawl, with detractors criticising it for making true combos rare and making it impossible to truly combo with moves that deal anything beyond very low knockback. This eliminates the comboing ability of characters that relied on higher knockback moves with faster mobility to combo (Captain Falcon being the most prominent example, with Jigglypuff also suffering severely), allows characters to survive longer than intended through momentum cancelling and significantly increases the chance of getting punished for landing a hit (Ganondorf's Dark Dive is a notable example of this, as its extremely weak knockback means that many characters can hit Ganondorf with a quick attack as soon as the move completes).
While some players argue in support of the mechanic, such as by stating that players should have to successfully read their opponent's actions to get successful followups and that opponents should have means of escaping combos, the majority of competitive players, even those who prefer Brawl over Melee, see the mechanic as a negative addition to the game. As such, those who mod the game usually remove the mechanic unless they intend to keep the vBrawl engine intact (as in Balanced Brawl), and all major mods except the aforementioned have removed the mechanic completely.
In Super Smash Bros. 4, hitstun has a very slightly altered formula, which essentially reduces overall hitstun by one frame from Melee and Brawl. On top of this, the ability to act out of hitstun is still present, with characters generally being able to act out of hitstun after 40 frames with an air dodge, or 45 frames with an aerial; these numbers increase proportionally to the amount of knockback taken (beginning to increase after sustaining more than 69 frames of hitstun to cancel with an aerial or more than 74 frames of hitstun to cancel with an air dodge), and become irrelevant at extreme knockback values. 
Compared to Brawl, the window for canceling hitstun occurs significantly later, making momentum canceling effectively impossible and allowing moves to KO earlier. Furthermore, air dodging to cancel hitstun actually marginally worsens horizontal endurance, as similarly to Brawl, a fighter is not able to begin opposing their momentum with directional inputs (e.g holding left) until towards the end of the air dodge animation. This was put in Brawl so that air dodging to cancel hitstun was not too powerful, however in Smash 4, the period in an air dodge in which a fighter is able to oppose their momentum almost always falls after when the hitstun would have ended anyway. Gravity and falling speeds were also increased, making true combos past low percentages possible again.
Because of the changes to hitstun, characters such as Sheik and Captain Falcon are indirectly buffed, as their comboing abilities from previous games have been restored, while Brawl newcomers such as Meta Knight and Zero Suit Samus also benefit significantly by gaining a similar or even greater combo potential. However, with the retained ability to air dodge or attack out of hitstun at higher percentages, gravity increasing knockback as in Brawl and falling speeds still being slower than in Melee, true combos at higher percentages are exclusive to a few characters (and even then, they must not be at high percentages themselves), making them less prevalent than in Melee and especially the original SSB.