Finchale Priory is a riverside ruin north of Durham, with enough stonework left to trace the church, cloister, and monastic buildings. Its setting on the Wear is the real pleasure: green, quiet, and far removed from the busy cathedral precinct.
It makes a good short excursion for travellers with a car or a taste for walks beyond the city. Durham Cathedral is grander, but Finchale gives you the more solitary side of medieval County Durham.