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From Citizendium, the Citizens' Compendium

A surface in mathematics has many different uses, the most common referring to a two-dimensional submanifold of three-dimensional Euclidean space, .

Any set of points composed of pieces topologically equivalent to a subset of a plane is a surface: this includes curved surfaces such as a paraboloid, infinite surfaces such as a plane, surfaces of limited extent such as the interior of a polygon, and surfaces with strange topology such as an infinitely long row of squares each separated by some distance, or the set of all points with rational coordinates in a plane. The extremities of a solid are made up of surfaces.

In Euclidean geometry:

A surface has length and breadth only. A surface that is flat is called a plane.