Dexterity is the character attribute that determines how limber your character is.

Diablo II

Dexterity affects your Defence, your Block with shields and your base damage with ranged weapons like bows or javelins.

Generally, except for Amazons, spare attribute points are better placed in Vitality to improve life. Character builds might suggest adding enough to dexterity to keep max block chance, max hit chance, or just enough to equip your items.

Diablo I

In Diablo I, Dexterity requirements on items were higher and all classes needed to keep it high. It was particularly important for Rogues. It affected your defence and chance to hit.