Magic wand is a common name for device allowing people cast spells they don't have. Technical name for them is spell catalyst. Spell catalyst doesn't need to be in shape of the wand: Tedd's magic watch are "technically a wand" or in other words also spell catalysts.
While magic wands can allow cast spells even to normal people, up to the point fireballs wands being used by Edward Verres as example of dangers of knowledge about magic, they are usually used by other magic users; only wizards can learn other people's spells, so if magic user need specific spells he don't have, or if group like DGB needs standard issue spells for every agent, they need to use wands with preset spells.
In addition to letting someone cast spells, spell catalysts can usually store energy, so caster doesn't need to use as much of their own energy to casting. Capacity of catalysts depends on size, but also material and quality of crafting. Tedd's watches are exception: they don't have any, only caster's energy and ambient energy are used with them.