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. SSB uses cuboids whereas SSBM and SSBB use spheres and sphere-like structures.
In SSB, hacks can enable hitbox visibility (with a few limitations), such as in this image. In SSBM, hitboxes can be seen using Debug Mode. There is no easy way to display hitboxes in-game in SSBB, 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).
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 SSB) 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. This characterisic is likely to prevent situations where a fast projectile may end up passing through a character in less than a frame without hitting them.
A trapbox is the hitbox left on the ground during Ness' Yoyo Glitch. As with all hitboxes, trapboxes are invisible. It is called a trapbox because this hitbox is not attached at all to any kind of attack but instead just sits on the ground or, in some cases, floats in the air.