Tangent space
The local coordinate frame on a surface — tangent, bitangent and normal — that normal maps are stored in so they work no matter how the surface is turned.
A normal map that stores directions in world space only works for a single orientation. Store the directions relative to the surface instead — z along the normal, x and y along the texture's u and v — and the same map can be pasted onto a curved, rotated or animated surface and still point "the right way". The dominant blue tint of a normal map is exactly the vector (0,0,1): straight out.
The basis is called TBN after its three axes and is computed per vertex from the gradients of positions and UV coordinates. One pitfall is handedness: mirrored UVs flip the bitangent, and forget the sign and a seam shows up along the mirror line. The term is borrowed straight from the tangent space of differential geometry, where it means all the tangent vectors at a point.