Is this correct?
Tensors are linear mappings between two coordinate systems on a manifold. The elements of that mapping (which include the different changes of bases at each point of the manifold) are governed by the components of the Jacobian. The Jacobian, the ratio of the volume elements of the two states – is itself a tensor.
(I expect this is elementary for those who know – I’d like to reach that state myself, so please feel free to expand).
You're not really correct about what a tensor is. Here's a more precise perspective that might help clear things up.
First remember that a vector $\mathbf v$ can be written
$$\mathbf v = \sum_i v_i \mathbf e_i$$
where $v_i$ are numbers called components and $\mathbf e_i$ are basis vectors.
A tensor is a generalization of a vector. By definition a rank $k$ tensor $\mathbf t$ can be written
$$\mathbf t=\sum_{i_1 \dots i_k} t_{i_1\dots i_k}\mathbf e_{i_1}\dots \mathbf e_{i_k}$$
An important thing to realize is that the components $t_{i_1\dots i_k}$ of a tensor only specify it uniquely if you know which basis $\mathbf e_{i_1}\dots \mathbf e_{i_k}$ you're working with!
You can smoothly assign a vector to every point on a manifold. When you do this you get a vector field.
Similarly when you assign a tensor to every point on a manifold you get a tensor field.
Because mathematicians and physicists are lazy they sometimes forget to use the word field. For example the metric tensor on a manifold should really be called the metric tensor field.
If we want to find out the components of a tensor field at each point on a manifold we must choose a set of coordinates to work in. It turns out that this provides an obvious basis $\mathbf{e}_{i_1}\dots \mathbf e_{i_k}$ for the tensors at each point on the manifold, allowing us to express them in components.
Sometimes we want to change variables though. This means that the associated basis for the tensor at each point on the manifold changes! Hence you get new components for the tensor according to tensor transformation rules.
The Jacobian is in fact an altogether more simple construct. We talk about it in the context of maps from $\mathbb{R}^n \to \mathbb{R}^n$. It is indeed a tensor field (since it's a matrix defined at every point on the manifold $\mathbb{R}^n)$. But that's not a particularly enlightening observation.
The Jacobian is useful because its determinant tells you how the unit volume element scales under the map. If you happen to have obtained your map $f$ by composing coordinate charts of a manifold $\mathbf{M}$ then the Jacobian determinant tells you how the volume elements differ in the two charts.
For a much fuller explanation, and a really good pedagogical introduction to tensors and volume forms I recommend Lee's Introduction to Smooth Manifolds.
