The Torelli map $\tau\colon M_g \to A_g$ sends a curve C to its Jacobian (along with the canonical principal polarization associated to C); see [this][1] question for a description which works for families. 

> Theorem (Torelli): If $\tau(C) \cong \tau(C')$, then $C \cong C'$.

If one prefers to work with coarse spaces (instead of stacks) it is okay to just say that $\tau$ is injective.

> Question: Is $\tau$ an immersion?

(One remark: $\tau$ isn't a closed immersion -- the closure of its image consists of products of Jacobians!)


  [1]: https://mathoverflow.net/questions/7505/are-jacobians-principally-polarized-over-non-algebraically-closed-fields/7513#7513