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 S

^{2} 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.