A vector is a line defined by length and direction, a normal in geometry is a line that is perpendicular to the object of your interest. Thus in 3D-geometry, a normal vector is a line that is perpendicular to your 3D-model's surface.
It is used for lighting calculations: angle of incidence = angle of reflection, the angle is between the light source the vertex point of the surface and the normal vector.
L | / \ n| / \ | /r \ | / \|/----------- Surface
L = Lightsource
n = normal vector
r = reflected light
The "normal map" texture modifies the angle of the normal vectors during the calculation of the lighting for each point, and thus can create the impression of more surface details. When the normal vectors are missing, the surface "absorbs" all light and appears black.
If you don't care about the theory, just right click on the NiTriShape in Nifskope and select "Mesh / Face Normals"

To see the normal vectors in Nifskope, select the "Normals" entry in the NiTriShapeData.
You'll also need tangents and binormals for good looking metallic surfaces, I won't go into too much detail here, just imagine the ------ line to be a sheet of squared paper that lies on the Text "Surface" which is the 3D model. The two axes of the paper would be the tangent and binormal vectors ( by definition the tangent is the line we actually can see in the 2D-sketch)
To add tangents and binormals in Nifskope set "BS Num UV Sets" in the NiTriShapeData to 4097
and then use Spells / Batch " Update All Tangent Spaces"