An Nasiriyah's main draw is its proximity to two UNESCO World Heritage sites: the ruins of Ur, 20 kilometres to the southeast, and the Mesopotamian Marshes to the east. Ur is easily the more celebrated — the ziggurat is one of the best-preserved in Iraq — but the marshes and the Marsh Arab villages offer a more unusual and in many ways more profound experience.
Within the city itself, the Nasiriyah Museum holds a good collection of objects from the surrounding archaeological sites, including Sumerian artefacts from Ur and Lagash. It is worth visiting before heading to the sites to understand what you are looking at.