I have computed the Alexander Polynomial through the skein relation but sources such as Wikipedia and nLab say:
The Alexander quandles are also important, since they can be used to compute the Alexander polynomial of a knot. [Both websites say exactly this]
However, take for example the quandle representation of the trefoil (pg 25):
$$a = b \triangleleft (a \triangleleft b)$$ $$b = (a \triangleleft b) \triangleleft (b \triangleleft (a \triangleleft b))$$
How do I use this information to compute the Alexander polynomial ($x^{-1} - 1 + x$) of the trefoil?
What we are looking for is a $\mathbb{Z}[t,t^{-1}]$-module $A$ to label the arcs of the trefoil that satisfies the quandle representation. In particular, $A$ will be a cyclic module, and the torsion of $A$ is the Alexander polynomial.
Using $a\triangleleft b=(1-t)a+tb$, the first equation $0=b\triangleleft (a\triangleleft b)-a$ is $0=(b-a)(1-t+t^2)$ and the second equation $0=(a\triangleleft b)\triangleleft (b\triangleleft (a\triangleleft b))-b$ is $0=(b-a)(t-1)(1-t+t^2)$.
The $b-a$ factor in each corresponds to the fact that no matter the choice of $A$, there is a trivial labeling of the Alexander quandle. The $1-t+t^2$ corresponds to the fact that $A=\mathbb{Z}[t,t^{-1}]/(1-t+t^2)$ has a non-trivial labeling. (In fact, one may assign $a,b$ to be anything in $A$.)