Maybe it's best illustrated by these two pictures (which sometimes say more than 1000 words):
The first picture represents stress, of which there are three different kinds, as you can see (therefore stress is a second order tensor quantity).
The second picture represents traction. The red $\tau$ vector is the component of the traction in the direction of the surface.
As you can read, in the definition of traction your book is talking about a force on a surface, while in the definition of stress force across a surface is mentioned.