Sasha Kleshchev's book "Linear and Projective Representations of Symmetric Groups" is the reference I'd suggest. Chapter 1 contains the connection with Young's lattice, and the subsequent chapters develop the functors that Ben described above. The second half of the book develops the theory for spin representations of symmetric groups which is an honest type B analogue (The functors $E_m$ and $F_m$ in Ben's answer satisfy the Serre relations for the Kac-Moody algebra of type $B_\infty$).

To add a little more detail to Ben's answer, the right level of generality to think about these questions is the affine Hecke algebra (either the degenerate or nondegenerate varieties). I'll describe the degenerate case:

Let $F$ be an algebraically closed field of characteristic $p$. As a vector space the (degenerate) affine Hecke algebra is a tensor product of a polynomial algebra with the group algebra of the symmetric group: $H_d=F[x_1,\ldots,x_d]\otimes FS_d$. Multiplication is defined so that each tensor summand is a subalgebra, and $H_d$ satisfies the mixed relations $s_ix_j=x_js_i$ for $j\neq i,i+1$ (here $s_i=(i,i+1)$), and $s_ix_i=x_{i+1}s_1-1$.

Note that in addition to being a subalgera, $FS_d$ is also a quotient of $H_d$ obtained by mapping $s_i\mapsto s_i$ and $x_1\mapsto 0$ (so that the $x_i$ map to Jucys-Murphy elements).

The polynomial subalgebra forms a maximal commutative subalgebra, so given a finite dimensional $H_d$-module $M$, we may decompose $$M=\bigoplus_{(a_1,\ldots,a_d)\in F^d}M_{(a_1,\ldots,a_d)},$$ where $$M_{(a_1,\ldots a_d)}=\lbrace m\in M|(x_i-a_i)^Nm=0,\mbox{ for }N\gg0\mbox{ and }i=1,\ldots,d \rbrace$$ is the generalized $(a_1,\ldots,a_d)$-eigenspace for the action of $x_1,\ldots,x_d$. Let $I=\mathbb{Z}1_F\subset F$ and $Rep_IH_d$ be the category of finite dimensional $H_d$-modules which are integral in the sense that if $M\in Rep_IH_d$, and $M_{(a_1,\ldots,a_d)}\neq 0$, then $(a_1,\ldots,a_d)\in I^d$.

Now, let $K_d=K_0(Rep_IH_d)$, and $K=\bigoplus_d K_d$. Then, the categorification statement is that parabolic induction and restriction give $K$ the structure of a bialgebra, and as such $$K\cong U_{\mathbb{Z}}(\mathfrak{n}).$$ In the above statement, $\mathfrak{n}\subset \mathfrak{g}$ is the maximal nilpotent subalgebra of the Kac-Moody algebra $\mathfrak{g}$ generated by the Chevalley generators $e_i$, where, if $char F=p$, then $\mathfrak{g}=\hat{sl}(p)$, and if $char F=0$, $\mathfrak{g}=\mathfrak{gl}(\infty)$. In both cases $U_{\mathbb{Z}}(\mathfrak{g})$ denotes the Kostant-Tits $\mathbb{Z}$-subalgebra of the universal enveloping algebra. Note here that the Chevalley generators are indexed by $I$.

Now, for each dominant integral weight $\Lambda=\sum_{i\in I}\lambda_i\Lambda_i$ ($\Lambda_i$ the fundamental dominant weights) for $\mathfrak{g}$, define the polynomial $$f_\Lambda=\prod_{i\in I}(x_1-i)^{\lambda_i}.$$ Then, the algebra $H_d^\Lambda=H_d/(H_d f_\Lambda H_d)$ is finite dimensional. In the case $\Lambda=\Lambda_0$, $H_d^\Lambda\cong FS_d$.

One can form $K_d(\Lambda)$ and $K(\Lambda)$ as above corresponding to the category $H_d^\Lambda-mod$. Then, the categorification statement is $$K(\Lambda)\cong V_{\mathbb{Z}}(\Lambda)$$ as $\mathfrak{g}$-modules, where $V(\Lambda)$ is the irreducible $\mathfrak{g}$-module of highest weight $\Lambda$ generated by a highest weight vector $v_+$, and $V_{\mathbb{Z}}(\Lambda)=U_\mathbb{Z}(\mathfrak{g})v_+$ is an admissible lattice. The action of the Chevalley generators on $K$ are analogues of the functors in Ben's answer. The action of the Weyl module corresponds to the action of $D=\sum_{i\in I}e_i$ and $U=\sum_{i\in I}f_i$ (in characteristic 0, this is defined in the completion $\mathfrak{a}(\infty)$ of $\mathfrak{gl}(\infty)$.

One can generalize this story to $\hat{\mathfrak{sl}}_\ell$ by working with the (nondegenerate) affine Hecke algebra $H_d(t)$, where $t$ is a primitive $\ell$-th root of unity. In this case, the finite dimensional quotients are Hecke algebras of complex reflection groups. The hyperoctohedral group corresponds to the highest weight $\Lambda=2\Lambda_0$. Then $V(\Lambda)$ is a level 2 representation, hence the central element acts by $2\cdot Id$ as in Sammy's comment).

In the second half of Kleshchev's book, the Hecke algebra is replaced by the so-called Hecke-Clifford (or Sergeev) algebra, and $\mathfrak{g}$ is of type $B_\infty$ or $A_{2\ell}^{(2)}$ depending on the ground field (or one can work in the non-degenerate case so that $\ell$ needn't be prime.

The algebras introduced by Khovanov-Lauda and Rouquier generalize this story to arbitrary symmetrizable Kac-Moody algebra. These algebras are graded, so one gets a categorification of the quantum group $U_q$, where $q$ keeps track of the grading . . .