<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 dual 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.

Argh, time for dinner, I'll come back soon.