In your setting $X$ is semi-stable means that its special fiber $X_s$ is a reduced divisor with normal crossings on $X$.
The link with Galois representations is very deep. In fact in general only one implication is known, namely if $X$ is semi-stable then its associated Galois representation is semi-stable. This was known as (a consequence of) the conjecture $C_{\mathrm{st}}$ of Fontaine and Jannsen. There are now at least three different proofs of this conjecture. One was given by the Japanese school (Hyodo, Kato, Tsuji), see Tsuji's survey in Astérisque 279. Another was given by Faltings using his theory of almost étale extensions. Recently Niziol gave another proof using $K$-theory.
The converse implication seems very difficult in general. For abelian varieties this was proved by Coleman-Iovita (Duke Math. 1999) and Breuil (Annals of Math. 2000). For curves this follows from Deligne-Mumford's theorem that a curve is semi-stable if and only if its Jacobian is semi-stable (Publ. Math. IHÉS 1969) (see Mathieu Romagny's article).
Faltings also has a result that the Galois representations associated to proper smooth schemes over $K = \operatorname{Frac}(R)$ are de Rham (and thus potentially semi-stable). So if we knew the converse implication in general, then we would deduce that every scheme is potentially semi-stable (in the sense that it acquires semi-stable reduction after a finite extension), but this is not known in general.