Strictly, 'tangent' means a vector that touches the surface at precisely one point. Not the surface itself. So in the particle setting context:
tangent
controls the starting force that is applied to the particle parallel to the emitter surface.
Normal
controls the force that is applied to the particle perpendicular to the emitter surface.
To put it another way,
tangent = 1, normal = 0 will make the particle emit 'sideways'.
tangent = 0, normal = 1 will make the particle emit 'out'.
tangent = 1, normal = 1 will make the particle emit at 45degrees to the surface.