One definition (#1) of the orientation of a manifold $X$ is a choice of orientation on $T_xX$ for all $x \in X$ which varies continuously with $x$. So an orientation on a manifold by definition corresponds to choosing orientations on $T_xX$ continuously for all $x \in X$.
Aside on orientations of vector spaces
(#1') Let $V$ be a real $n$-dimensional vector space, and $(v_1, \dots, v_n)$ and $(v_1', \dots, v_n')$ be bases of $V$. Then $v_i'= \sum_{j=1}^n A_{ij}v_j$ for $A \in \mathrm{GL}(n, \mathbb{R})$. Now define an equivalence relation on bases of $V$ by $(v_1, \dots, v_n) \sim (v_1', \dots, v_n')$ if and only if $\det A > 0$. Then an orientation of $V$ is a choice of equivalence class $[ v_1, \dots, v_n ]$.
(#2') There is a second notion of orientations of vector spaces. Let $V$ be as above. Then $\bigwedge^n V \cong \mathbb{R}$ and we have $0 \neq v_1 \wedge \cdots \wedge v_n \in \bigwedge^n V$. By what we've said, $\bigwedge^n V \setminus 0 \cong (-\infty, 0) \cup (0, \infty)$ has two connected components.
It is a fact that if $(v_1, \dots, v_n)$ and $(v_1', \dots, v_n')$ are bases of $V$ with $v_i'= \sum_{j=1}^n A_{ij}v_j$ then
$$ v_1' \wedge \cdots \wedge v_n' = \det A \cdot v_1 \wedge \cdots \wedge v_n . $$
So define an equivalence relation $(v_1, \dots, v_n) \sim (v_1', \dots, v_n')$ if and only if $ v_1' \wedge \cdots \wedge v_n'$ and $v_1 \wedge \cdots \wedge v_n $ lie in the same connected component of $\bigwedge^n V \setminus 0$.
You may ask how the definition of orientation of a manifold that I've given above and the definition you've given in terms of charts are equivalent.
Firstly, as we've said in (#2') above, one can interpret an orientation on a vector space $V$ as choosing a connected component of $\bigwedge^n V \setminus 0$. Applying this definition of orientations of vector spaces to the vector spaces $T_xX$ gives a second definition for the orientation of a manifold:
#2: An orientation of an $n$-manifold $X$ is an equivalence class $[ \omega ]$ of non-vanishing top forms $\omega \in \Omega^n(X)$ where $\omega$ is equivalent to $\omega'$ if and only if $\omega'=f \circ \omega$ for a smooth function $f : X \to (0 , \infty)$.
We finally come to the third definition (your definition) of the orientation of a manifold:
#3: We first define oriented charts. Let $X$ be an $n$-manifold with orientation $[ \omega ]$. Let $(U, \phi)$ be a chart on $X$. We call $(U, \phi)$ oriented if $\phi^*(\omega) = f \cdot dx_1 \wedge \cdots \wedge dx_n$ where $f : U \to \mathbb{R}$ and $f >0$. we can find an atlas $\mathcal{A} = \{ (U_i, \phi_i) \mid i \in I \}$ for $X$ consisting of only oriented charts. Call $\mathcal{A}$ an oriented atlas. For two such charts on $X$ with local coordinates $(x_1, \dots, x_n)$, $(y_1, \dots, y_n)$ we have $\det \left( \frac{ \partial y_i}{\partial x_j} \right)_{i, j=1}^n > 0$ on the overlap. Then we can define an oriented manifold to be a manifold with oriented atlas.
Hopefully this chain of definitions sufficiently links together the way of thinking about orientations of manifolds via charts and the way of thinking about orientations of manifolds as orientations of their tangent spaces.
For $n \not= 4$, a smooth manifold homeomorphic to $\mathbf R^n$ is diffeomorphic to $\mathbf R^n$, so $\mathbf R^n$ as a topological manifold has just one smooth structure. The story is totally different for $n = 4$: look up exotic $\mathbf R^4$. There are $28$ different smooth structures on a $7$-dimensional sphere. That is, if you consider smooth manifolds that are homeomorphic to the smooth manifold $S^7$ but not necessarily diffeomorphic to it, there are $28$ examples. Look up exotic spheres.
In describing such phenomena, you could choose to talk about manifolds that are homeomorphic but not diffeomorphic or you could choose to talk about different smooth structures on a specific topological manifold. The choice is up to you.
Best Answer
In the 19th century, three basic models of geometry were known to mathematicians:
It turns out that all of these geometries, even when developed purely axiomatically, can be understood better once we study curved surfaces. For example, the Gauss-Bonnet theorem is one of these theorems that in its basic form connects the sum of the interior angles of a triangle on a curved surface with the curvature of the surface. Obviously, $2$-dimensional surfaces like sphere, torus, hyperboloids, etc. can be understood by us well because they can be embedded in $\mathbb{R}^3$ and we can use our tools developed for calculus in $\mathbb{R}^3$ to study them.
In 1818, Gauss was assigned to carry out a geodetic survey in the Kingdom of Hanover. During this time, he made measurements and invented tools like Heliotrope to do the job. It says that this job helped him formulate Theorema Egregium which he was very fond of it and was a starting point for developing calculus on surfaces.
Theorema Egregium (which means the Remarkable Theorem in Latin) says that an ideal ant (a two dimensional being) can understand the curvature of the surface it lives on by making local measurements. The significance of this is that this enables us humans to study higher dimensional surfaces by developing a calculus on the surface locally, instead of doing calculus on the ambient space. This also has physical implications. We humans are three dimensional beings, living in a universe which is probably four dimensional. Naturally, we are eager to understand the geometry of the universe around us. Theorema Egregium tells us that we are not completely hopeless and we can determine some local geometric properties of our universe just by local measurements.
For further reference, please read about Gauss–Codazzi equations, Riemannian geometry and General Relativity.