A hitbox or collision bubble (sometimes hitbubble) is the main structure for how attacks are executed in most fighting games. Attacks have one or more hitboxes associated with them, and when these hitboxes overlap with a target's damagable area (sometimes called their hurtbox or hurtbubbles), the attack is considered a hit. Hitboxes are invisible and usually, though not always, have the same shape as the attack's animation.
The two most common forms of hitbox shapes in video games are cuboids and spheres. Neither is obviously better than the other - cuboids can more easily form long or thin attacks or targets, while spheres are easier to calculate collision detection for. Smash 64 uses cuboids whereas Melee and Brawl use spheres and sphere-like structures known as capsules (cylinders with spheres on the ends).
In Smash 64, hacks can enable hitbox visibility (with a few limitations), such as in this image. In Melee, hitboxes can be seen using Debug Mode. There is no easy way to display hitboxes in-game in Brawl, though a common technique is to use hacks to place Super Scope shot visuals where the hitboxes are (which is significantly more involved a process and generally fails for any projectile or item).
Should multiple hitboxes of a single move connect with the opponent, only one of them will count. The order of precedence is known as the "hitbox stack" - hitboxes higher in the stack will override ones lower should they both hit at the same time. Mechanically, this is denoted by giving each hitbox an ID number, and lower numbers indicate higher precedence (higher stack position). For example, Marth's down aerial in Melee features its tipper hitbox on top of the stack, so hitting with both a tipper and non-tipper hitbox will result in the tipper counting (even if the majority of the opponent was hit by the other hitboxes). By contrast, the move in Brawl has the tipper on the bottom, so the tipper can only hit if none of the other hitboxes do. As a result, even through the sizes and positions of the hitboxes did not change much between games, the tipper is simple to hit with in Melee and impractical in Brawl.
Hitboxes that are large and/or distanced from the attacking character are often called disjointed, which is often an advantage, as they can connect from a surprising distance while keeping the attacker away from retaliation. Examples of disjointed hitboxes include most sword and hammer attacks, as well as the up tilts of Kirby (in Smash 64) and Snake.
An interesting property of hitboxes in the Smash Bros. series is that under most conditions, they occupy not only the space where they currently are, but also the space where they were one frame ago as well as all the space in between (in a straight line, regardless of what the animation might look like during intervening subframes). This characteristic is likely to prevent situations where a fast projectile may end up passing through a character in less than a frame without hitting them. On the other hand, hurtboxes do not stretch over "interframe space" in this way, so a fast-moving object (such as Fox during his Illusion) may be able to move through a slow or stationary hitbox without being damaged. In addition, certain hitboxes in Brawl use the stretch effect to create a very long hitbox, for example in Zero Laser and Aura Storm, instead of stretching over interframe space. It is unclear how the hitbox-stretching mechanism operates in Brawl when time is slowed via training mode or other effects, as Brawl renders subframes, whereas Melee and SSB64 simply reduce the overall framerate.
It is a common misconception that, like some other fighting games, hitboxes and hurtboxes only operate in two dimensions. In actuality, they interact in all three dimensions equally, and there are many situations where certain animations can bring some of a character's hurtboxes outside the line of fire for many attacks (such as the end of King Dedede's sidestep).
 Hitbox properties
A trapbox is the hitbox left on the ground during Ness' Yoyo Glitch. It is called a trapbox because this hitbox is not attached at all to any kind of attack but instead just sits wherever it was when the glitch occured. Using a new attack will overwrite the hitbox with the new attack's properties and relocate it to the new attack, which due to the interframe stretching effect will cause it to hit everything between it and Ness.