Vertex tangents and normals- what are they and how do I calculate them?

It is my basic understanding that on a surface (or plane), a normal is perpendicular to it, and a tangent is (somehow) parallel to it. Firstoff, how is being parallel to the plane tangent? If the normal would be a line, it would intersect the plane at only one point, so wouldn't that make the normal the tangent? Secondly, how is it possible to have a normal or tangent to a vertex? A vertex is a single point in space, nothing can be parallel to it or perpendicular to it, so how does that work? Or am I misunderstood and possibly the tangents and normals are calculated on a per-polygon basis? Please, I am trying to understand these concepts in 3d graphics, and I would like to know these things. I would also like to know how to calculate these given a single quad or triangle, if at all possible. Thanks in advance to whoever can clear this up!