A tangent vector is a vector, which is tangent to a curve, surface, or, more generally, a manifold at a certain point. If you have a curve c(t), parametrized by t, its tangent vector is just c'(t), where ' means the derivative. A manifold, of course, generally has many tangent vectors. In fact, it has a tangent vector space at its each point. Locally any manifold looks like its tangent vector space, but in general not globally. More concretely, if you have a sphere S2 in 3-dimensional space, if you look at it closely enough, it looks flat, but as you zoom farther away, the curvature becomes apparent.

A tangent vector can be mathematically defined as a directional derivative in some direction on the manifold. Tangent vectors are very important in differential geometry.

amsaarel is correct in (his? her?) writeup, however, I felt that it needed to be expanded:
If you have an n+1 dimensional manifold M of paramaters t_1, t_2, ..., t_n, then you have an n dimensional vector space which has all the tangent vectors in it. To find the vector with greatest slope (dM, the complete differential) use the vector differential operator del, defined as:
del:=<@/@t_1,@/@t_2,@/@t_3,...,@/@t_n>
where @/@x is the partial with respect to x operator. del(M) will yield a vector in the vector space, and by dotting del(M) with unit vectors, directional derivatives can be found in those directions. By using a multidimensional cross product between all n vectors found, the normal to the vector space can be found, this an equation for the vector space can be found.
I now feel tempted to write "The proof is left as an exercise to the reader" here since it annoys me so!

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