I consider only finite-dimensional Hilbert spaces.
We know that pure states on $\mathcal{B}(\mathcal{H})$ are exactly the vector states or in terms on density matrices, the rank one projections.
My question is now, is there something similar to determine purity of states on subalgebras of $\mathcal{B}(\mathcal{H})$?
Yes, since you are in finite dimension. Any $A\subset B(H)$ is of the form $$A=\bigoplus_{j=1}^m M_{k_j}(\mathbb C).$$ It is not hard to check that every state is of the form $\varphi:x\longmapsto \operatorname{Tr}(bx)$ for some $b\in A_+$ with $\operatorname{Tr}(b)=1$ (and, thus $\|b\|\leq1$).
We have $b=\sum_{r=1}^s b_rq_r$, where $q_r$ are pairwise orthogonal rank-one projections. The above conditions on $b$ give $b_r\geq0$ for all $r$, and $\sum_r b_r=1$. If $s\geq2$, we can write $$ \varphi(x)=\sum_r b_r \operatorname{Tr}(q_rx) $$ and we get $\varphi $ written as a convex combination of the states $x\longmapsto \operatorname{Tr}(q_rx)$. So, if $\varphi$ is pure, we have $s=1$. That is, $b$ is a rank-one projection.
In summary, the pure states of $A$ are precisely the maps $x\longmapsto \operatorname{Tr}(qx)$, where $q\in A$ is a rank-one projection.
It would remain to prove that every rank-one projection gives rise to a pure state. You can see a proof here.