TITLE: Geometrical meaning of semi-stable reduction?
QUESTION [20 upvotes]: So let $R$ be a discrete valuation ring and let $X$ be a scheme which is proper and flat over $R$. Let $X_s$ denote the special fiber of $X$. 
So intuitively, when somebody says that a curve $X$ is semistable I kind of equate this in my mind with the property that $X_s$ has only ordinary double points as singularities.
Q1: So in general (i.e. in higher dimension) what is the geometrical meaning for a scheme
to be semistable?  
On the Galois representation side we have a very precise definition of what semistable means using Fontaine's ring $\mathbf{B}_{st}$. 
Q2 If there is a precise answer to Q1, is there a good reference (more on the intuitive side than on the technical side) where one shows the equivalence (under suitable assumptions) that the geometrical definition coincides with the galois representation one?

REPLY [15 votes]: Just to elaborate on Francois Brunault's answer: it is not true in general that having semistable etale cohomology implies having semistable reduction (just as it is not true that having crystalline etale cohomology implies having good reduction).  So the implication only goes in one direction.
Added in response to comment below: E.g. if $E$ is an elliptic curve over $\mathbb Q_p$
with good reduction, and $P$ is an $E$-torsor with no $\mathbb Q_p$-rational point, then $P$ will not have good reduction.  It will not have semi-stable reduction either, since it has
potentially good reduction (it obtains a rational point over some extension of $\mathbb Q_p$, and hence becomes isomorphic to $E$ over that same extension).  The etale cohomology ($\ell$-adic or $p$-adic) of $P$ coincides with that of $E$, and so $P$ is an example of a variety over $\mathbb Q_p$ with semistable (indeed crystalline) Galois action on its $p$-adic etale cohomology, which does not have semistable reduction.