A flinch is the reflex all characters make when their current action is interrupted by an attack. The interrupted action is replaced with a short animation of being hit. The knockback dealt affects how strong the flinch is - weak attacks incur only a few frames of flinching, while stronger ones can deal about three times as much (and for some characters, even more if their legs are struck), but those strong enough to cause tumbling skip the flinch animation altogether. As a result, attacks that do not deal any knockback will not cause flinching. However, windboxes are able to deal flinchless knockback to fighters.
As flinching is directly related to knockback received, reducing knockback through a sticker or Equipment, increasing weight through a Super Mushroom or Metal Box, or using a move with armor, will also make it more difficult (or impossible) to flinch. Metal Mario and Giga Bowser (in Melee only) are a few examples of characters who don't flinch when hit with low knockback attacks, due to a combination of their weight, along with the inclusion of knockback resistance, heavy armor, and/or knockback taken multipliers. In Super Smash Bros. 4, Bowser will not flinch from attacks that deal less than 19 units of knockback. However, all fighters will flinch if they take more than 0 units of knockback, and they take more damage (for damage-based heavy armor) or knockback (for knockback-based heavy armor) than what their heavy armor can sustain.
Flinches can make characters vulnerable to combos, and the lag from them can sometimes guarantee a followup with a stronger attack or a grab. A prime example is the jab grab in Super Smash Bros. (Captain Falcon can do a similar technique in Brawl), or the jab-to-up smash combo that works with Fox in both Super Smash Bros. and Melee.
Flinching while charging an attack or special move will cause the charge to be lost, regardless of whether any charge had been saved previously. Mewtwo does not lose its projectile's charge if it flinches, though this is only the case for when it is about to fire the projectile, not while it is actually charging.