When Solved, Archaeology Fragments form Artifacts. Solved Artifacts can be found in Archaeology's Completed Artifacts tab.
Most artifacts are common. These give you a little bit of lore or flavor text and an item you can sell for a small profit (presumably to a museum!) The profit increases as the value of the artifact increases. You can estimate this by the number of fragments needed to finish the artifact. You will only find cheaper artifacts at low level, but you can find cheap and valuable artifacts at higher level (the reason for this is we want players to be able to find all the artifacts if they want to). You won't find a second copy of a common object until you have found all the artifacts of that race. If you get stuck at a certain skill level of archaeology or character level, you may find an artifact more than once until you reach the next tier.
Some artifacts are rare. These always make a blue or purple item. Many of these have no in-game power and are toys or for flavor. Some of them are actual weapons and armor. The latter items are all bound to account. They aren't heirlooms in that they don't scale, but you can pass them around. So if you are level 80 and you find a level 60 axe, you can always have another character use it when they are level 60. You will never get a rare artifact more than once.