Tensor product of modules over an arbitrary algebra is not always defined? If $V,W$ are finite dimensional vector spaces over a field $F$ and $G$ a group acting on $V$ and $W$ (from right) then $G$ also acts on $V\otimes W$ as follows: fix a basis of $V\otimes W$; for a basis element $v\otimes w$, define action of $g\in G$ on it by 
$$(v\otimes w).g=v.g\otimes w.g.$$
Then extend this action linearly on space $V\otimes W$, and then extend linearly (the domain of action) from $G$ to $F[G]$. 
I. Martin Isaacs (in Character theory book) points out (p. 48) that 

if $A$ is an arbitrary algebra, then it is not generally possible to define the tensor product of $A$-modules $V$ and $W$.

I didn't understand this statement; where the problem arises in defining tensor product of modules over an algebra? Can one illustrate it with some simple example? 
 A: Well, the definition you gave doesn't even make sense for an arbitrary algebra.  It involves defining $$(v\otimes w).g=v.g\otimes w.g$$ when $g\in G$, and then extending to all elements of $F[G]$ by linearity.  If $A$ is an arbitrary algebra, what is "$G$" in this definition?
You might say, just take "$G$" to be some basis of $A$ as a vector space.  However, this runs into trouble with associativity.  For $g,h\in G$, we have $$((v\otimes w).g).h=(v.g\otimes w.g).h=v.(gh)\otimes w.(gh).$$  On the other hand, to compute $v\otimes w.(gh)$, we first need to write $gh$ as a linear combination of elements of $G$, say $gh=\sum a_ix_i$ for $x_i\in G$ and $a_i\in F$.  We then have $$v\otimes w.(gh)=\sum a_iv\otimes w.x_i=\sum a_iv.x_i\otimes w.x_i.$$ There is no reason for this sum to be equal to $v.(gh)\otimes w.(gh)$.  In the case $A=F[G]$, though, $gh$ is guaranteed to already be an element of $G$, so the sum $\sum a_ix_i$ has only one term, namely $x_1=gh$ and $a_1=1$.  As a result, associativity does work.
In any case, even if the module structure defined in this way ended up being associative, it would depend on the choice of a basis $G$.  So the module structure on $V\otimes W$ is not intrinsic to the algebra $A$: it involves additional structure on $A$ besides its algebra structure (namely, a choice of basis).  In general, it turns out that the natural additional structure needed to define an $A$-module structure on a tensor product of $A$-modules is that $A$ should be a bialgebra, not just an algebra.
