Of the six stats Dexterity is a sort of "fused stat," representing a character's ability to move not only precisely and gracefully, as the name would suggest, but quickly and reactively. It is associated with many Skills, more than both the other physical stats combined (though this is mostly because no or almost no skills are Con-based), like Sleight of Hand, Use Rope, Acrobatics, Balance, Tumble, Disable Device, and, perhaps most famously, Stealth. Unusually, many of these skills are actually quite useful in combat, such as Acrobatics and Tumble to move over difficult terrain and prevent free attacks from enemies as characters pass, or making Stealth tests to hide and set up sneak attacks or escape sticky situations. It also has a high impact on combat by virtue of offering benefits to Armor Class and Initiative, with the former representing a character's ability to avoid attacks and the latter a character's quick reflexes letting them strike first, and is the stat that governs ranged attacks. Modern editions also allow it to affect saving throws or defenses to avoid area-of-effect spells and traps; in 3.X it is the ability score used to calculate Reflex saves and even in 5e it's one of the three most common saves in the game.
Like many a "speed" stat, Dexterity is usually, pound for pound, one of the most powerful and generally-useful stats in the game. Ranged weapon users and skill monkeys rely on it as their God Stat, all lightly armored classes need it to make up for their lack of armor, casters need it to go first and hit enemies while they're all bunched up together, and even the more heavily armored classes usually want the highest dex modifier that their armor will allow them to benefit from.
In particular, 5e gave it a massive boost. Traditionally, it was somewhat-difficult for a Dexterity-based fighting character to add their stat bonus to damage rolls, ensuring a Strength-based character could at least usually hit harder. In 5e, any ranged weapon or any weapon with the "finesse" keyword can just flat-out use Dexterity for attack and damage rolls instead. And while previous editions made relying on Dexterity for defense a bit of a risk, via incorporating mechanics that could strip away the benefits of Dexterity to AC by catching a character "flat-footed" via surprise, restraints, or skillful maneuvering, 5e does not. All this on top of Dexterity still offering some of the only combat-useful skills in the game, more skills overall than any other non-mental stat, and more combat benefits overall via its bonuses to various combat functions. The only consolation given to Strength characters is that at least wearing heavy armor removes the AC penalty for low Dexterity, which was not the case previously.
Dexterity is also a "physical" stat, since it comes from a character's body. It is the traditional stat-of-choice for the sneaky rogue, the swashbuckling versions of the bard, and the dodgy, unarmored monk. However, because of its wide range of benefits, most classes benefit from having a high dex score. SAD classes will want to pour all of their leftover stat points into constitution and dexterity after maxing out their god stat. Only heavily armored MAD melee classes like the Paladin and Psychic Warrior find themselves forced to treat dexterity as a dump stat