A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface.
A hypersurface may have singularities, which are the common zeros, if any, of the defining polynomial and its partial derivatives. In particular, a real algebraic hypersurface is not necessarily a manifold.
Hypersurfaces have some specific properties that are not shared with other algebraic varieties.
is a real hypersurface without any real point, which is defined over the rational numbers. It has no rational point, but has many points that are rational over the Gaussian rationals.
These two processes projective completion and restriction to an affine subspace are inverse one to the other. Therefore, an affine hypersurface and its projective completion have essentially the same properties, and are often considered as two points-of-view for the same hypersurface.