In all Golden Sun games, an Artifact is any "special" item (as opposed to any "common item"). They typically have unique, strong, or - in the case of equipment - overwhelming effects. Artifacts can do many things, and their effect is told in their item description. Unlike common items, these can be bought and sold individually through a shopkeeper's "Artifacts" (ほりだしもの, Bargains?) menu. If an Artifact is either sold or dropped, it can be repurchased from the appropriate shopkeeper.
Some Artifacts require an actual command to be used. Water of Life, which instantly revives a downed ally to full health, is one such Artifact. Others include any of the Rings, any other strong healing item such as a Potion, and certain pieces of equipment, such as the Blessed Mace and Captain's Axe.
Unique to weapons, some Artifacts will randomly unleash certain effects when a regular attack is performed. The actual Unleash is different for each weapon; some inflict status conditions, such as the Darksword's Acheron Grief. Others might lower stats, like the Hestia Blade's Crucible Fire. Others, such as the Sol Blade's Megiddo, simply do tremendous amounts of damage to the opponent.
Note that as of Dark Dawn unleashes are no longer unique to artifact weapons. All weapons, except for rusty items prior to reforging, now have up to four unleashes which only become available provided the wielder is experienced enough with the specific class of weapon.
Many Artifacts - primarily defensive artifacts, but also some weapons, such as the Gaia Blade - just have a passive effect that happens as long as they are equipped. They range from recovering small amounts of PP and HP to increases in stats other than Defense.