What are fallow fields? I'm a city gal who heard the term at a 4-H fair and just read it in Anna Karenina.

Fallow means uncultivated or unplanted, or, in a more cerebral sense, untrained or inactive. A fallow field is land that has been plowed and not seeded for one or more growing seasons for a purpose - to kill weeds or to enrich the soil, for example.

In Leo Tolstoy's Anna Karenina, Konstantin Levin finds farming, manual work, and his relationship with the peasants a source of satisfaction:

When they came out of the woods, all [Konstantin Levin's] attention was engrossed by the view of the fallow land on the upland, in parts yellow with grass, in parts trampled and checkered with furrows, in parts dotted with ridges of dung, and in parts even ploughed.