What does the Tate module of an elliptic curve tell us?

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 $\mathbb C$.

There are many, many ways in which the knowledge of the various Tate modules gives you information about the elliptic curve. For example:

Theorem(Neron-Ogg-Shafarevich): Let $E$ be an elliptic curve over a number field $K$. Let $p$ be a prime of $\mathbb{Q}$. Then, $E$ has good reduction at $\mathfrak{p}$, for $p\nmid\mathfrak{p}$, if and only if the representation $\rho_p:G_K\to \text{GL}_2(\mathbb{Z}_p)$ is unramified at $\mathfrak{p}$.

Here $G_K$ is the absolute Galois group of $K$, and $\rho_\mathfrak{p}$ is any of the representations of the form

$$G_{K_\mathfrak{p}}\to G_K\to \text{GL}_2(\mathbb{Z}_p)$$

where $G_K\to \text{GL}_2(\mathbb{Z}_p)\cong \text{GL}_2(T_p E)$ is the representation coming from the action of $G_K$ on $T_p E$. Also, being unramified means that this representation's kernel contains the inertia group $I_\mathfrak{p}$.

In fact, you can recover almost all the local $L$-factors of an elliptic curve over a number field from its Tate modules (and its Galois actions), which, by a hard theorem of Tate/Faltings, actually determines $E$ up to $\mathbb{Q}$-isogeny. So, in a very real sense, the Tate modules (and their associated Galois representations) determine everything about the elliptic curve, at least over a number field.

Let's start over the complex numbers, so that $E$ is an elliptic curve over $\mathbb C$. Then the natural map $\mathrm{End}(E) \to \mathrm{End}(H^1(X,\mathbb C))$ is injective. Suppose now that you have an elliptic curve over $\mathbb F_p$. What could the analogue of this statement be?

If $E$ is an elliptic curve over an arbitrary field $k$, and $\ell$ is a prime number invertible in $k$, then the Tate module $\mathrm{V}_\ell(E) = T_{\ell} E \otimes \mathbb Q$ is a replacement for $H^1(X,\mathbb C)$ (in some sense). For instance, for all prime numbers $\ell$ invertible in $k$, the natural map $\mathrm{End}(E) \to \mathrm{End}_{\mathbb Q_\ell}( V_\ell E)$ is injective.

Roughly speaking, the Tate module is the (dual of the) $\ell$-adic realization of the "motive" of $E$. In fact, it is more than just a $\mathbb Q_\ell$-vector space (of dimension two) as it carries a natural action of the absolute Galois group $G_k = Gal(\bar k/k)$ of $k$.

If $k$ is finitely generated over its prime field, then the natural map

$\mathrm{End}(E) \otimes \mathbb Q_\ell \to \mathrm{End}_{\mathbb Q_\ell[G_k]}( V_\ell E)$ is an isomorphism. This is due to Tate, Zarhin and Faltings.

So in arithmetic situations (e.g., number fields or finite fields), the Tate module tells you essentially everything about the motive.

Sometimes properties of the motive imply similar properties of the underlying variety. The criterion of Neron-Ogg-Shafarevich (see Alex Youcis's answer) is an important example of this phenomenon. In fact, the criterion of Neron-Ogg-Shafarevich tells you that an elliptic curve $E$ over a discretely valued field $K$ has good reduction over $O_K$ if and only if there exists a prime number $\ell$ with $\ell \in k^\ast$ such that the $\ell$-adic Tate module (read "$\ell$-adic realization of the motive of $E$") has good reduction over $O_K$ (in the sense that it is unramified as a Galois representation).

Another application comes from the before-mentioned theorem of Tate, Zarhin and Faltings. If $E_1$ and $E_2$ are elliptic curves over a finitely generated field $k$ and $\ell \in k^\ast$, then any $G_k$-equivariant morphism $V_\ell E_1 \to V_\ell E_2$ is induced by an element of $\mathrm{Hom}(E_1,E_2)\otimes \mathbb Q_\ell$. Therefore, the elliptic curves $E_1 $ and $E_2$ are $k$-isogenous if and only if their $\ell$-adic Tate modules (read ``$\ell$-adic realizations of their motives") are isomorphic as Galois representations.

If $k$ is finitely generated of characteristic zero, you can also read of the system of Tate modules $\{V_\ell(E)\}_{\ell}$ whether your curve has complex multiplication.

Similar statements hold for abelian varieties.

Before you try to understand the above facts in full detail, I recommend you follow guy_in_seoul's advice and develop your intuition for the Tate module by studying elliptic curves over $\mathbb C$.