Coffee cake, in American usage, nearly always refers to a sweet cake intended to be served as a breakfast or brunch dish, or with a mid-day coffee or tea service. It is analogous to the British teacake.
American coffee cakes are very often single-layer cakes, often chemically leavened with baking powder, and frequently flavored with cinnamon, nutmeg, and fruit. They may often include a streusel topping. They are likely derived from the Streuselkuchen, brought by German immigrants as part of the kaffeeklatsch custom.
For a cake that is actually flavored with coffee, a more common term is mocha cake or espresso cake.