I've recently met Jacobian varieties, in the context of elliptic curves and modular curves. I'm still very new to them and not super comfortable with the machinery, so please forgive me any mistakes here. One convenient thing to do is to let $\varphi: X_0(N) \to E$ be a modular parameterization. Then letting $J(\cdot)$ stand for taking the Jacobian, $\varphi$ ought to transfer to a map $\varphi_*: J(X_0(N)) \to J(E)$ which is both a morphism of varieties and a group homomorphism, right? The group structure of the Jacobian lets you do things that aren't really possible to do directly with $\varphi$ on $X_0(N)$ and since $E \cong J(E)$ by $P \mapsto [P-O]$, it's easy to move from $J(E)$ to $E$ itself.
One question is about what this induced map ought to look like. Given $D = \sum a_i P_i$ a degree-$0$ divisor, is $\varphi_* D = \sum a_i \varphi(P_i)$ (which is again degree $0$)? (And you have to check this makes sense when modding out the principal divisors.) I'm a bit unsure of what's going on because don't you usually induce maps on divisors going the other way? Furthermore, in the case of $\varphi:X_0(N) \to E$, what's the relationship between "summing up points in divisors" and "summing up points according to the group law on $E$"? I do know there is a lemma (from Silverman's Arithmetic of Elliptic Curves) that says a divisor on $E$ is principal if and only if it is degree $0$ and the points in it sum to $O$ under $E$'s group law.
One other fact I know is that in the case of curves over $\mathbb{C}$ (i.e. Riemann surfaces), there is a very concrete description of the Jacobian, as periods modulo homology, giving a $g$-dimensional torus. Most of my exposure to Jacobians is through Diamond and Shurman's "A First Course in Modular Forms."
I apologize for a question that's a bit all over the place, but what's the "main thing" I should take away from Jacobian varieties? How do their basic properties work, and what is the underlying reason they are useful for doing number theory and arithmetic/algebraic geometry?