Priority
From SmashWiki, the Super Smash Bros. wiki
For example, if two Marios punch each other at the same time, the attacks will "cling" (that is, both attacks will be stopped) and neither will take damage. This is because the attacks have the same priority; this can also happen if the priority of the two attacks are similar but not completely equal. But if Mario's weak punch were to collide with Yoshi's forward smash, then Yoshi's attack (with its higher priority) will go right through Mario's attack, damaging him without Yoshi taking damage.
If two characters attack each other in the air at the same time, then they can hit each other regardless of priority (unless one of the two attacks comes out faster), though if one or both of the attack are a Special, priority still decide who gets hit. If one of the characters is on the ground and the other is in the air, priority decides who gets hit, but if it's the same priority, both trade hits. Priority only affects grounded characters.
Generally, more damaging attacks have higher priority. For example, a forward smash will almost always trump a jab (except, in Brawl, for Captain Falcon's forward smash which, despite being powerful, has extremely low priority). The main exception to this is grabbing: a grab always has more priority than an attack. When the attacks "collide", the grab overrides the attack and stops the grabbing character from flying back but still takes the damage (this does not work with Olimar since his Pikmin grab the characters and not Olimar himself). The grabbing characters get no frames of knockback resistance simply because the grab has higher priority and therefore stops the characters from flinching. Also, sword-wielding characters generally have high priority in their sword attacks, and if the opponent out-prioritize the sword, the character still doesn't get hit (unless they hit the character instead of the sword) as the sword hitbox is disjointed.
Generally, having poor priority is a major issue in a character since it makes approaches very hard, and is the main reason why Captain Falcon and Ganondorf are low tier. Likewise, having high priority is a major advantage, which is a major reason why characters like Meta Knight, Marth, and Snake are high tier.
