Let $X/K$ be a variety over a global field $K$. When (and why) is the Galois representation $H^i_{et}(X \times_K \bar{K}, \mathbf{Q}_\ell)$ unramified at a place $v$ of $K$?
I guess this is true if $X$ has a model smooth (or regular?) over $v$ by using base change theorems.
Same question for Hodge-Tate/de Rham/crystalline/semistable.
[Added: Just to set the scene, this answer is discussing the $p$-adic etale cohomology of a variety $X$ over a finite extension of $\mathbb Q_p$ (or, a little more generally, a finite extension of the fraction field of the Witt vectors of a perfect field of char. $p$).]
Crystalline implies semi-stable implies de Rham implies Hodge-Tate. The cohomology is crystalline if $X$ is proper with a smooth model, is semi-stable if $X$ is proper with a semi-stable model, and is de Rham always. (In the proper case, the last statement is a theorem of Tsuji; in the general case, of Kisin. The crystalline and semi-stable cases are due, in various degrees of generality, to Fontaine--Messing, Hyodo--Kato, Faltings, and Tsuji.)
Added: It might help to note that de Rham and potentially semi-stable coincide (as discussed in the comments below, Berger reduced this to a result in the theory of $p$-adic differential equations, which was then proved independently by Andre, Kedlaya, and Mebkhout). Also, unramified is a very strong condition in this setting ($p$-adic etale cohomology of varieties over $p$-adic fields), which essentially never holds unless we are looking just at $H^0$, and if the geometrically connected components of $X$ are defined over an unramified extension.
By asking "when is it ramified," you might be also be asking whether there are any other conditions under which it's ramified, i.e. a converse to Emerton's answer. For abelian varieties, the converse is true, and this is known as N\'eron-Ogg-Shafarevich. There are $p$-adic Hodge theory versions as well.
However, in general, a variety with Galois representation unramified at $v$ need not have good reduction at $v$. In all cases where this has been proven, one relates the Galois representation in question to the representations of auxiliary varieties that do have good reduction at $v$. As a simple example, there are smooth projective curves with bad reduction at $v$ but whose Jacobian has good reduction at $v$.
One might conjecture that any Galois representation coming from a variety actually comes in some fashion from varieties with good reduction over $v$. More precisely, any motive over a global field $K$ whose associated $\ell$-adic Galois representation is unramified at a finite place $v$ (whose residue characteristic $p$ is not $\ell$) is the base change of a motive (of smooth proper varieties) over $\mathcal{O}_{K,v}$, the local ring at $v$. One can make a similar conjecture for $\ell=p$ if one assumes the representation is crystalline. One can even make a modification by defining a category of motives using semistable varieties. I don't know if this is written down anywhere, but I and other graduate students I know have wondered this.
As far as I remember, the representation is always potentially semistable (this means that it is Hodge-Tate and de Rham?). It is crystalline when $X$ has a smooth model; it is semistable if it has a log-smooth model (I am not sure that this is the right name).
The question about unramified is still open. (Sorry for using the answer for bumping up.)
