I think at least historically the Jacobian is related to the function theory over a curve which was one the main areas of research back in the 19th century. In that time given a compact Riemann surface $X$ over $\mathbb{C}$, the question was to understand the behavior of holomorphis and mermorphic functions on this curve. If we have to effective divisors $D$ and $E$ on $X$, when is $D-E$ the divisors of zeros and poles of a meromorphic function $f$ on $X$? Let the genus of $X$ be $g>0$. Then there are $g$ basis elements of the vector space of differential forms on $X$. The clever solution that Abel proposed for this question was this: let $\omega_{1}$,...,$\omega_{g}$ be the generators of  $\Omega(X)$, the space of holomorphic differentials of $X$. Given a path $\gamma$ in $X$, the set $L=\{(\int_{\gamma}\omega_{1},...,\int_{\gamma}\omega_{g})\}$ is additive in $\mathbb{C}^{g}\cong\Omega(X)$ because of additivity property of integrals and in fact is a lattice. Therefore we can quotient out and get a group $\mathbb{C}^{g}/L$. We also get a map $ A:X\rightarrow J(X)$ by choosing a base point and $p_{0}$ and sending each point $p\in X$ to $(\int_{p_{0}}^{p}\omega_{1},...,\int_{p_{0}}^{p}\omega_{g})$ mod $L$. Abel realized that two divisors $D$ and $E$ (viewed as a collection of points on $X$) are linearly equivalent if and only if they have the same image under the map $A$. Note that the map $A:X\rightarrow J(X)$ in itself is a very interesting map: we have constructed an almost natural holomorphic map form $X$ to a variety that has a structure of a group. In the first glance it is not at all clear that we can have such a map. The second funny property is that this map is not injective if and only if $X\cong \mathbb{P}^{1}$.