<i> Abstract Definition </i>.
Let $Rep(G)$ be the abelian category of smooth complex representations of our $p$-adic group $G$. The Bernstein center is the endomorphism ring $\mathfrak Z(G)$ of the identity functor of $Rep(G)$. So it acts on any smooth representation, and this action commutes whith any $G$-morphism.

<i> As a projective limit </i>. Let $H$ be a compact open subgroup. Letting $\mathfrak Z(G)$ act on the permutation representation $\mathbb C[G/H]$ gives a morphim to the center $Z(G,H)$ of the Hecke algebra $\mathcal H(G,H)$. This yields an isomorphism $\mathfrak Z(G) \simeq \lim\limits_{\leftarrow H} Z(G,H)$ where transition maps are given by applying idempotents.


<i> Geometric realization</i>. (here "geometric" is in the sense of Trace formulas, i.e. on the side of harmonic analysis). $\mathfrak Z(G)$ acts on the regular representations $C^\infty_c(G)$. The pairing $(z,f)\mapsto z.f(1)$ embeds $\mathfrak Z(G)$ as a set of distributions on $G$. The image is the convolution algebra of "essentially compact invariant distributions".

<i>Spectral realization </i>. By Schur's lemma (which holds in this context), $\mathfrak Z(G)$ acts on any irreducible representation $\pi$ via  a character $\theta_\pi:\mathfrak Z(G)\longrightarrow \mathbb C$. This is sometimes called the "infinitesimal character" of $\pi$, by analogy with the archimedean situation, although there is nothing "infinitesimal" here. We get in this way a realization of $\mathfrak Z(G)$ as an algebra of continuous functions on the smooth dual $\hat G$ of $G$ equipped with the Fell topology. 

Sofar, nothing deep. Now, two major achievements in the representation theory of p-adic groups are 
<ol> <li>the Bernstein theorem which describes the spectral realization explicitly,</li> 
<li>the Harish Chandra Plancherel formula which provides a link between both realizations.</li>
</ol>


Let me try to describe Bernstein's result. Bernstein first splits the category $Rep(G)$ as a (infinite) product of indecomposable abelian subcategories (called "blocks"). Accordingly, the smooth dual decomposes into infinitely many connected components, and the center decomposes as an infinite product of rings.

The simplest example is that of a compact $G$ (eg the kernel of the norm map in a division algebra). In this case, $\hat G$ is discrete and the center is a product of copies of $\mathbb C$ indexed by the set of classes of irreps.
The next example is that of a compact-mod-center $G$ (eg the unit group of a division algebra). In this case there is an action of the group $\Psi(G)$ of unramified characters of $G$ on $\hat G$. Note that $\Psi(G)$ is naturally an algebraic torus over $\mathbb C$ because $G$ mod its maximal compact subgroup is a free abelian group of finite type. Now, connected components are the orbits of $\Psi(G)$, and the topology is the homogeneous space topology. In particular each such orbit carries a natural structure of an algebraic variety over $\mathbb C$. Finally, $\mathfrak Z(G)$ is the direct product of the ring of regular functions on these orbits.

Let us go to the general case. 
Assume first that $G$ is semisimple. Then each supercuspidal representation gives an isolated point in the smooth dual $\hat G$, because such representations are both projective and injective objects. Therefore, denoting by $Cusp(G)$ the set of (isom classes of) supercuspidal irreps, the ring $\mathbb C^{Cusp(G)}$ is a factor of $\mathfrak Z(G)$. If $G$ is reductive, then $Cusp(G)$ is still open and closed in $\hat G$, but is not discrete. As above, $\Psi(G)$ acts on $Cusp(G)$ and the latter is the disjoint union of orbits under $\Psi(G)$ with the natural quotient topology. The corresponding product of ring of regular functions on each orbits is then a factor of $\mathfrak Z(G)$, the "cuspidal" part $\mathfrak Z(G)_{cusp}$ of $\mathfrak Z(G)$.

Now remains the crucial step of describing the non-cuspidal part of the center. Bernstein uses parabolic induction from Levi subgroups. One quick way of stating the final result is :
$$ \mathfrak Z(G)\simeq \left( \prod_M \mathfrak Z(M)_{cusp} \right)^G$$
Here the product is over all Levi subgroup and $G$ acts by conjugacy. In order to get something less frightening, one can fix a maximal split torus $T$, restrict the product to those $M$'s that contain $T$ (finitely many) and take $N_G(T)$-invariants.

In particular, connected components are labeled by conjugacy classes of $\Psi(M)$-orbits
 in $Cusp(M)$. 
Suppose $G$ is split for simplicity. A particularly interesting component is that which corresponds to the $\Psi(T)$ orbit of the trivial character of $T$. Its contribution to 
$\mathfrak  Z(G)$ is isomorphic to $\mathbb C [X(T)]^W$. 
This component contains the unramified representations and the action of $\mathbb C[X(T)]^W$ on each such representation is given by its Satake parameter. In fact the corresponding connected component of $\hat G$ is the set of irreps that have non-trivial invariant under an Iwahori-subgroup, and one recovers the fact that the center of the Hecke-Iwahori algebra identifies with $\mathbb C[X(T)]^W$.
Finally, note that this ringlooks like a "group version" of the center of an envelopping algebra, so the analogy with the archimedean context is even deeper than expected.