Wikipedia says a volume form on a differentiable manifold is a nowhere-vanishing top-dimensionial form.
And Guillemin and Pollack says $f: V \to U$ is a diffeomorphim of two open sets in $\mathbb{R}^k$ and $\omega = dx_1 \wedge \cdots \wedge dx_k$ is the volume form.
So, can I understand volume form as the top-dimensional form with coefficient $1$?
And as an exapmple, if $\omega$ is the volume form on $T^2$, then $\int_{T^2} \omega = \operatorname{vol}(T^2)?$
The space of top forms is the space of sections of a real vector bundle of dimension 1 over your manifold. A volume form is a non-vanishing section of this bundle. So as Potato says, if there is one volume form (I.e. if your manifold is orientable), then there are many. You can, however, distinguish a unique volume form if your manifold is oriented and Riemannian: You then require the form to take the value 1 on oriented orthonormal bases at each point.