What does the Tate module of an elliptic curve tell us? I started studying elliptic curves, and I see that it is rather common to take the Tate module of an elliptic curve (or, of the Jacobian of a higher genus curve).
I'm having a hard time isolating the benefits of this construction. Can you give an example (or claim) explaining how the Tate module sheds light on some properties of the curve?

You should begin by carefully understanding, in the case that $E = \mathbb C/\Lambda$ is an elliptic curve over the complex numbers, the canonical isomorphism between the $\ell$-adic Tate module and $\mathbb Z_{\ell} \otimes_{\mathbb Z} \Lambda$ (or, what is the same, the inverse limit $\varprojlim_n \Lambda/\ell^n\Lambda$).  
Once you've understood that, you could read the proofs in Silverman classifying the possible endomorphism rings for elliptic curves.  Silverman's proofs work in arbitrary characteristic, and use Tate modules.  But you could try to imitate them
for elliptic curves over $\mathbb C$, using the lattice $\Lambda$ directly as a tool.  The arguments then become quite a bit simpler.  Comparing these simple arguments with the Tate module arguments should further build up your intuition.
The next step is to learn the proof of the Hasse--Weil theorem counting points on elliptic curves over finite fields, and the philosophy of thinking of its an application of the Lefschetez fixed point theorem for the Frobenius endomorphism --- with the Tate module playing the role of $H_1$.   If you recall that in the complex case the lattice $\Lambda$ is canonically identified with $H_1(E,\mathbb Z)$, this will add yet more intuition.
Summary: Tate modules substitute for the lattice $\Lambda$ which plays such an important role in the study of elliptic curves over $\mathbb C$.
