I know the jacobian from multivariate calculus, i.e. not in the context of differential geometry.
Differential geometry gives a nice abstract language for differentiation. I find the concept of a "tangent space" particularly clarifying.
I don't know how the jacobian fits in this picture. Is there a differential geometry version of the jacobian?
Certainly. Take a smooth map between manifolds $F:N\rightarrow M$, and charts $(U,\varphi)=(U, x^1,\cdots, x^n)$ on $N$ and $(V,\psi)=(V, y^1,\cdots, y^m)$ on $M$, so that $F(U)\subset V$. Define the coordinates of $F$ relative to the chart $(V,\psi)$ by $F^i=y^i\circ F.$ Then, the matrix $[\partial F^i/\partial x^j]$ is the Jacobian matrix of $F$ relative to the charts. If $M$ and $N$ have the same dimension, then you can also take the determinant. You can easily check that this coincides with the Jacobian that you know from calculus if you take standard coordinate charts.
Here, partial derivatives are defined via $$\frac{\partial f}{\partial x^i}=\frac{\partial (f\circ\varphi^{-1})}{\partial r^i}\circ \varphi,$$ where $r^i$ are the Euclidean coordinate functions.
Typically, we take charts around a point $p\in N$ and $F(p)\in M$, in which case the matrix representation is $[\partial F^i/\partial x^j(p)]$, with respect to the bases induces by the local coordinates at $p$ and $F(p)$. This ends up being the matrix representation of the differential/pushforward $F_{*,p}$, which is an induced map on the tangent spaces.