A vector field on a manifold is a (continuous, differentiable) map $X: M \to TM$ such that $X(p) \in T_pM$ for each $p \in M$. The tangent space $T_pM$ has a basis.
Take $M = \mathbf{S}^2$. For each $p \in \mathbf{S}^2$ there is a basis for $T_p\mathbf{S}^2$ consisting of two vectors $v_p$ and $w_p$. Hence (I think) vector fields on $\mathbf{S}^2$ are exactly the maps of the form $p \mapsto f(p)v_p + g(p)w_p$ for some (continuous, differentiable) maps $f, g: \mathbf{S}^2 \to \mathbf{R}$.
So, considering this, why doesn't, say, the vector field $p \mapsto v_p$ give a counterexample to the Hairy Ball theorem, i.e. a nonvanishing vector field? Since $v_p$ and $w_p$ form a basis for $T_p\mathbf{S}^2$, $v_p$ is never zero, right?
As you can see, I am confused. Thanks in advance for your help.