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.