$\def\Alb{\text{Alb}}\def\Pic{\text{Pic}}\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}\def\cO{\mathcal{O}}$There are two abelian varieties associated to a smooth projective connected $n$-fold $X$: The Albanese variety $\Alb(X)$ and the Picard variety $\Pic^0(X)$. The Albanese has a natural map $X \to \Alb(X)$ which induces an isomorphism $H^1(\Alb(X)) \to H^1(X)$; the Picard variety parametrizes $(n-1)$-cycles of degree $0$ up to rational equivalence. For a curve, we have $\Alb(X) \cong \Pic^0(X)$, and we call both of these the Jacobian. So we have to understand three things: The Albanese variety, the Picard variety, and the isomorphism.
This causes the ambiguity about whether this question is straightforward or not: It is pretty straightforward that $H^1(X) \cong H^1(\Alb(X))$. But the description in terms of $0$-cycles is a description of $\Pic^0(X)$. What I'm describing in this answer is a modern (or at least, 20-th century version) of the Abel-Jacobi theorem which gives a criterion for determining when two $0$-cycles on a curve are rationally equivalent in terms of integrating holomorphic $1$-forms.
The Albanese variety: Let's start by just thinking about a smooth connected manifold $M$. We get a map $H_1(M, \ZZ) \longrightarrow H^1_{DR}(M, \RR)^{\vee}$ by sending a $1$-cycle $\gamma$ to the linear functional $\omega \mapsto \int_{\gamma} \omega$. The kernel is the torsion part of $H_1(M, \ZZ)$, so the image is the torsion free quotient $H_1(M, \ZZ)_{tf}$. Define $\Alb(M)$ to be $H^1_{DR}(M, \RR)^{\vee}/H_1(M, \ZZ)_{tf}$. Topologically, this is just a torus whose dimension is the first betti number of $M$. We clearly have a natural isomorphism $H_1(\Alb(M), \ZZ) \cong H_1(M, \ZZ)_{tf}$ and so $H^1(M, \ZZ) = \text{Hom}(H_1(M, \ZZ)_{tf}, \ZZ) \cong H^1(M, \ZZ)$.
We can get a map $M \to \Alb(M)$ as follows: Choose a vector space $V$ of closed $1$-forms on $M$ which maps isomorphically to $H^1_{DR}(M)$, and choose a base point $x_0 \in M$. For any $x \in M$, choose a path $\beta$ from $x_0$ to $x$. Then $\eta \mapsto \int_{\beta} \eta$ is a linear functional on $V$. If $\beta'$ is a different path from $x_0$ to $x$, then $\beta' - \beta = \gamma$ for a $1$-cycle $\gamma$, so $\int_{\beta'} \eta = \int_{\beta} \eta + \int_{\gamma} \eta$. In other words, replacing $\beta$ by $\beta'$ changes the linear functional $\int_{\beta} (\cdot)$ by an element of $H_1(X, \ZZ)$. So the class of $\int_{\beta} (\cdot)$ in $V^{\vee}/H_1(X, \ZZ)$ depends only on $x$, and we get a map $M \to V^{\vee}/H_1(X, \ZZ)$ sending $x$ to $\int_{\beta} (\cdot)$. Since we choose $V$ to be isomrophic to $H^1_{DR}(M)$, we get a map $M \to \Alb(M)$, and it is easy to check that this map induces the isomorphism $H^1(\Alb(M), \ZZ) \cong H^1(M, \ZZ)$.
Everything becomes more canonical if $M$, which I'll now call $X$, is a connected compact Kahler manifold, for example, a smooth connected complex variety. In that case, Hodge theory tells us that $H^1(X, \CC) = H^{10}(X) \oplus H^{01}(X)$, where $H^{10}(X)$ is the global holomorphic $1$-forms, $H^{10}(X) = H^0(X, \Omega^1)$. Concretely, this isomorphism says that we can take our $V$ to be the real parts and the imaginary parts of holomorphic $1$-forms (or, equivalently, we can take $V$ to be the real harmonic $1$-forms). Thus, $\Alb(X) = H^0(X, \Omega^1)^{\vee}/H_1(X, \ZZ)_{tf}$. Now $H^0(X, \Omega^1)^{\vee}$ becomes a complex vector space, so $\Alb(X)$ is not merely a real manifold but a complex manifold.
The Picard variety Let $X$ be a smooth complex manifold. Divisors on $X$, modulo rational equivalence, are the same as line bundles on $X$, and are the same as classes in $H^1(X, \cO^{\ast})$. (This should be in most algebraic geometry textbooks.) I am going to work in the analytic world here, so $\cO$ is the sheaf of holomorphic functions; $\cO^{\ast}$ is the sheaf of non-vanishing holomorphic functions and my topology is the analytic topology.
We have the exponential sequence of sheaves $0 \to \underline{\ZZ} \overset{2 \pi i}{\longrightarrow} \cO \overset{\exp}{\longrightarrow} \cO^{\ast} \to 1$, where $\underline{\ZZ}$ is locally constant $\ZZ$-valued functions. So we get a long exact sequence of cohomology which includes $$H^1(X, \underline{\ZZ}) \longrightarrow H^1(X, \cO) \longrightarrow H^1(X, \cO^{\ast}) \longrightarrow H^2(X, \ZZ).$$ The kernel of the map to $H^2(X, \ZZ)$ are called the cycles of degree $0$ and denoted $\Pic^0(X)$, so we have $\Pic^0(X) \cong H^1(X, \cO) / H^1(X, \underline{\ZZ})$.
Again, things are nicer if $X$ is connected compact Kahler. Then $H^1(X, \cO) = H^{01}(X)$ and the map $H^1(X, \ZZ) \to H^1(X, \cO)$ is the composition of $H^1(X, 2 \pi i \ZZ) \subset H^1(X, \CC) \to H^{01}(X)$ where the second map is the projection onto the second summand of the Hodge decomposition. (I'm going to start getting sloppy about dropping the $2 \pi i$.) In particular, it follows from Hodge theory that the image of $H^1(X, \ZZ)$ is a discrete, cocompact lattice in $H^{1}(X, \cO)$, so the quotient $H^1(X, \cO) / H^1(X, \underline{\ZZ})$ is a compact complex manifold.
The case of curves So far, we have two abelian varieties: $$\Alb(X) = H^0(X, \Omega^1)^{\vee}/H_1(X, \ZZ)_{tf} = H^{10}(X)^{\vee}/H_1(X, \ZZ)_{tf}$$ and $$\Pic(X) = H^1(X, \cO)/H^1(X, \ZZ) = H^{01}(X)/H^1(X, \ZZ).$$ But, if $X$ is a curve, then Poincare duality gives an isomorphism $H_1(X, \ZZ) \cong H^1(X, \ZZ)$ and Serre duality gives an isomorphism $H^{10}(X)^{\vee} \cong H^{01}(X)$. (In fact, Serre duality is just the Poincare duality pairing restricted to the two Hodge summands of $H^1(X, \CC)$.)
After checking enough compatibility of diagrams, this gives an isomorphism $\Alb(X) \cong \Pic^0(X)$, and gives that the map $X \to \Alb(X)$ we defined by integration (using the base point $x_0$) matches the map $X \to \Pic^0(X)$ sending $x$ to the divisor $[x]-[x_0]$.