Volume forms and volume of a smooth manifold Choose a volume form $\omega$ on $M$, oriented manifold. For every $F\in C^{\infty}_c(M)$, we define
$$
\int_M F:=\int_M F\omega
$$
where in the right hand term $M$ is taken wit positive orientation w.r.t. $\omega$, that is
$$
\omega=\lambda_p dx_1\wedge\dots\wedge dx_n
$$
for every $p$, and $\lambda_p>0$.
Let $M$ be a $n$-dimensional compact manifold and let $\omega\in\Omega^n(M)$ be a volume form. Define the volume of M as
$$
Vol(M):=\int_M 1=\int_M\omega
$$
Now I'm asking myself if the Volume is well-defined, i.e. if it doesn't depend on the choice of $\omega$.
I would say that it depends on the choice of $\omega$.
Infact, I know that $\omega$ would be, locally, something of the kind
$$
(\omega)_p=F_p(dx_1\wedge\dots\wedge dx_n)_p
$$
where $x_1,\dots,x_n$ are the coordinates induced by a local chart and $F\in C^{\infty}(M)$.
So $\omega$ is not unique. 
Probably I haven't learnt well this topic.
Remember that a volume form for a $n$-dimensional manifold $M$ is a line section of the canonical bundle $\Lambda^nT^*M$, i.e. a non-vanishing $n$-form.
 How can I prove that the volume of a compact manifold always positive?


Volume is non-unique. Therefore, the smooth structure of a smooth, compact manifold is not enough for the manifold to have a canonical volume.
What can happen is, that if your manifold has a richer structure, then you can associate a canonical volume to that richer structure. For example, if you have a Riemannian manifold $(M,g)$, then the differential form whose expression in a local chart, $(U,x)$, is $$ \omega=\sqrt{|\det g|}dx^1\wedge...\wedge dx^n $$ is a canonical volume form, and then the volume of $M$ as a Riemannian manifold is $$\int_M \omega. $$
Or if $(M,\omega)$ is a symplectic manifold, with $\omega$ being the symplectic form, then $\Omega=\omega^{\wedge n}$ is a canonical volume form, and the volume of $M$ as a symplectic manifold is $$ \int_M \Omega. $$
However, in either case, $M$ by itself has no meaningful volume. Volume isan extra structure you either choose, or a different, but chosen structure (like a riemannian metric, or a symplectic form) chooses for you.
