I am not familiar with the mathematical literature on various algebras, but I've briefly come across $q$-deformations a couple times in my work.
The main example I've seen has been the $q$-deformation of $SU(2)$, with algebra
$$[S^\pm, S^z] = \pm S^\pm$$ $$[S^+, S^-] = \frac{q^{2 S^z} - q^{-2S^z}}{q-q^{-1}}$$ To find representations of this algebra, my procedure is to note that the two-dimensional representation with $S^+ = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} $, $S^- = (S^+)^\dagger $, $S^z = \begin{pmatrix} 1/2 & 0 \\ 0 & -1/2 \end{pmatrix}$ naturally give rise to higher-dimensional representations through tensor products. Namely, we can use the "coproduct" $\Delta(S^+) = S^+ \otimes q^{S^z} + q^{-S^z} \otimes S^+$ to build representations on higher-dimensional spaces, and then break those up into irreducible representations. In my understanding, the commutation relations for the $q$-deformed algebra are chosen so as to ensure that the coproduct above (or simple variants thereof) indeed yields valid tensor product representations.
I have heard mention of "elliptic quantum groups", which have two parameters, $q$ and $p$. What are the commutation relations of the elliptic quantum group associated with $SU(2)$, and what is its associated coproduct? I fear my understanding of quantum groups may be "not even wrong," so please feel free to disabuse me of any notions in the comments.