What term is used for metamorphic rock with banded texture?
You may be referring to foliation which is the layering within a metamorphic rock. Foliation occurs when there is a strong pressure applied to the rock in one direction, just like when you put your hands flat together and push. When this pressure is applied, certain minerals, like mica and chlorite, grow with their long direction perpendicular to the pressure. This make the rock appear with layering and/or banding.