The tangent space [tex]T_pM[/tex] of the manifold M at point p can be defined as the vector space spanned by the basis vectors
[tex]\frac{\partial}{\partial x^\mu}\bigg\lvert_p[/tex]
where x is a coordinate system (a chart). (There is also a coordinate independent definition, but it will result in the same quantities being called tangent vectors, so it's equvalent to this one).
Let F denote the set of smooth ([tex]C^\infty[/tex]) functions from M into R (the set of real numbers). The vectors in the tangent spaces are linear functions from F into R
In the physics literature, a vector field is sometimes defined as a function that takes each point p in a subset U of M to a vector in the tangent space at p. In the mathematics literature the definition is a little more complicated, but it's still pretty close to the sloppy physicist's definition.
I'm not 100% sure that I remember the mathematical definition 100% correctly, but I think this is at least very close to it:
The tangent bundle TM of the manifold M is defined by
[tex]TM=\big\{(p,v)p\in M, v\in \bigcup_{q\in M}T_qM\big\}[/tex]
The function [tex]\pi:TM\rightarrow M[/tex] defined by
[tex]\pi(p,v)=p[/tex]
is called the projection.
A vector field is a section of the tangent bundle. A section is a function [tex]X:M\rightarrow TM[/tex], such that
[tex]\pi(X(p))=p[/tex]
