Cities are a kind of Building and Terrain. They are key to pretty much all battles and appear in almost every map in every game. They provide War Funds for repairs and the production of new units. Cities also offer defensive cover (3 stars) for Land Units and like all buildings do not inhibit movement for any unit. Allied cities will also refill rounds and fuel for free, as well as replenishing 2 HP at a cost, to those units.
Orange Star City
Orange Star city during capture animation
In Fog of War buildings do not show the color of their owner unless in vision of your units. A player must especially keep this in mind for cities as they are usually the most abundant building as well as being heavily fought over. It is a significant advantage if you can predict what cities your opponent could have captured when temporarily repelled from a part of the map. It also provides you with a way to make a rough estimate of how high the income of your enemy is.
In Advance Wars: Days of Ruin (Dark Conflict for non-Americans), cities, like all properties, allow you to see up to 2 squares away.