Stress is the general term for any force acting upon an area. It can be subdivided into normal stress, in which the stress acts perpendicular to the surface and shear stress, which acts parallel to the surface. If we now look further into normal stress the force can either be "pulling" on the surface which is then called tensile stress or it can "push" on the surface which is then called compressive stress or pressure.

Pressure is more to do with fluids, and always act perpendicular to the surface. Stress is more to do with solids, and it can act perpendicular (normal stress) or parallel (shear stress) to the surface.