The Poincare-Hopf Index Theorem states that:
Theorem: The index of a vector field with finitely many zeros on a compact, oriented manifold is the same as the Euler characteristic of the manifold.
But what is the definition of Index of a point or vector field? I would be appreciate if everybody provide a good reference or answer.
For every isolated zero of a vector field, the nearby points are nonzero, therefore for a small sphere around the zero, we may assign a unit vector. This is therefore a map from $S^n\to S^n.$ The index of this zero of the vector field is the degree of this map.
The index of the entire vector field is the sum of the indices of all the zeros.
For example, the vector field $v=x\hat{i}+y\hat{j}$ in $\mathbb{R}^2$ has a zero at $(0,0).$ At a point $(r\cos\theta,r\sin\theta)$ near the origin, the unit vector is $(\cos\theta,\sin\theta).$ This is a map from $S^1\to S^1$ of degree one.
Or the vector field $(x^2-y^2)\hat{i} + (2xy)\hat{j},$ which also has a zero at the origin. A point $(r\cos\theta,r\sin\theta)$ near the origin carries a vector $(r\cos2\theta,r\sin2\theta),$ whose unit vector is $(\cos 2\theta,\sin 2\theta).$ This is a map $S^1\to S^1$ of degree two (it wraps the circle twice).