In all three 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 are individually buyable and sellable one at a time through shopkeeper "Artifacts" menus. If you either sell an Artifact or drop it, you can buy it back from any shopkeeper that has an Artifact menu.
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.
In the case of pieces of equipment other than weapons (although Cursed weapons and the Gaia Blade could technically be considered in this category), some Artifacts just have an additional effect that acts as soon as they are equipped. They range from recovering small amounts of PP and HP to increases in stats other than Defense.