What ingredient or method causes a mixture to be a bread versus cake?
Multiple factors give cake and bread its distinct attributes:


Fat/sugar content: a bread is generally very lean. Most traditional breads do not have any fat or sugar in it at all. Cake will generally have a high fat content from either butter, oil, or milk. Breads will taste "drier" than cake because it lacks this fat content. Cake will also have a large content of sugar compared to bread. Often case the sugar is creamed with the butter to give the batter some of its structure. The more air that is incorporated into the sugar/butter mixture the "lighter" the cake will be.
Leavening: Cakes will be leavened using baking soda and/or baking powder. This type of chemical leavening is very quick and will result in lots of little holes within the dough resulting in uniform "soft" texture. Bread on the other hand is primarily leavened using yeast and requires time for the yeast to eat the simple sugars in the dough and produce carbon dioxide which creates the bigger and uneven holes in the bread.
Preparation of batter/dough: Cake will very often use AP or Cake flour which are low in gluten. These cake recipes will generally ask you to "mix until just incorporated." This is to insure that the batter is evenly mixed but not over mixed. This is to prevent gluten formation. Bread on the other hand want this gluten formation so it uses a flour with high gluten content such as bread flour. The kneading and working of the bread dough will align these gluten strands and give the bread the dough structures that make it chewy and "bready." 


Note that breads like banana bread, and zucchini bread fall under the category of quickbreads which  is actually more a cake than a bread.