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 re-actively. It is associated with many Skills, more than both the other physical stats combined, 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.
Like many a "speed" stat, Dexterity is usually, pound for pound, the most powerful and generally-useful stat in the game. No other stat incurs more penalties for dumping, and no other stat offers more benefits for focusing on. I mean, just read that paragraph above! Compare to Strength and weep.
In particular, 5e gave it a massive boost. Traditionally, it was somewhat-difficult for a Dexterity-based fighting character to add their score to damage rolls, at least meaning a Strength-based character would 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 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 sop to Strength characters is that at least wearing heavier armors removes the AC penalty for low Dexterity, which did not use to be the case.
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 no other stat offers more benefits for a high score or harsher penalties for a low one, literally any class in the game benefits from high Dexterity, whether it's spellcasters who can't wear armor, enjoy high-impact spells that function better at higher initiative, and sometimes need it to make ranged attacks with their magic, or just any fighting class character hoping to specialize in ranged weaponry and/or go earlier in the turn order.