All characters can use wands in DnD 5e, unless the magic item specifies that it requires attunement by a spellcaster. Wands that are magic items with charges of spells or other effects can be used by any class in some cases. But in other cases, they can only be attuned to by a spellcaster.
For example, the Wand of Fireballs requires attunement by a spellcaster:
While the Wand of Magic Missiles requires neither attunement nor a spellcaster user:
If a magic wand states that a “prerequisite is to be a spellcaster, a creature qualifies if it can cast at least one spell using its traits or features, not using a magic item or the like.” (DMG 137-8). So, if you have any spells, even from a feat like Magic Initiate or a racial feature like Tiefling’s Infernal Legacy, you are considered a spellcaster.
Most magic wands with higher-level effects, spells, and rarity levels require attunement by a spellcaster. A character can be attuned to no more than three magic items at a time under normal conditions (DMG 138).
Wand as an Arcane Focus or Druidic Focus
A wand can be used as a spellcasting focus by Sorcerers, Warlocks, Wizards, Druids, and Rangers. A spellcasting focus can be used in place of the material components specified for a spell, unless the material component has a cost requirement and/or the material component is consumed.
Sorcerers, Warlocks, and Wizards can use a wand as an arcane focus. Druids and Rangers can use a yew wand as a druidic focus.
These wands cost 10 gp apiece and do not have additional magical properties, like holding charges of spells. They are only useful as spellcasting foci for the classes that can use them as such.