Need a primer on dice first? Start here.
Wondering what kind of dice you need for Dungeons and Dragons? The answer depends somewhat on what kind of character you're playing.
Everyone Needs:
No matter what kind of character you're playing, you'll want a set of standard RPG Dice (a D4, D6, D8, D10, D1%, D12 and D20).
In addition, all characters can gain Advantage (a bonus where you use the higher of two D20 rolls) and Disadvantage (a penalty where you use the lower of two D20 rolls) so you may want an additional D20.
Depending on your class, there are additional dice you will want:
Melee Classes (Barbarian, Fighter, Monk, Rogue)
Most melee characters will want additional dice corresponding to their weapon's damage. Additionally, Rogues will want additional D6s for their Sneak Attack class feature.
Divine Magical Classes (Cleric, Druid, Paladin, Ranger)
Clerics, Paladins and Druids will want extra D4s and D8s for their healing spells. Paladins and Rangers often use melee attacks so will likely also want extra dice for their weapon's damage.
Arcane Magical Classes (Wizard, Sorcerer, Warlock, Bard)
Wizards, Sorcerers and Warlocks will want a variety of different dice depending on the spells they typically cast - D4s for Magic Missile, D6s for Fireball and Disintegrate and D8s for Cone of Cold are the 3 most common spells that use large numbers of dice.
Bards have an ability called Bardic Inspiration which allows them to assign bonus dice to other players. As such, they will want Bardic Inspiration dice equal to their charisma bonus. This starts as a D6 but increases at higher levels, so the die needed will change throughout the character's life.