Background.
According to Raymond A. Ryan, in his book Introduction to Tensor Products of Banach Spaces, a Banach ideal $\mathcal{J}$ is an assignment to each pair of Banach spaces $X$ and $Y$ a linear subspace $\mathcal{J}(X,Y)$ of $\mathcal{L}(X,Y)$ together with a norm $\rho_{X,Y}:\mathcal{J}(X,Y)\to[0,\infty)$, such that the following properties are satisfied.
(I1) For every $x^*\in X^*$ and $y\in Y$, the one-dimensional operator $x^*\otimes y$ belongs to $\mathcal{J}(X,Y)$, and $\rho_{X,Y}(x^*\otimes y)\leq\|x^*\|\|y\|$.
(I2) For any pair of Banach spaces $W$ and $Z$, if $T\in\mathcal{J}(X,Y)$ with $A\in\mathcal{L}(W,X)$ and $B\in\mathcal{L}(Y,Z)$, then $BTA\in\mathcal{J}(W,Z)$ and $\rho_{W,Z}(BTA)\leq\|B\|\rho_{X,Y}(T)\|A\|$.
(I3) The linear space $\mathcal{J}(X,Y)$ is complete under $\rho_{X,Y}$, i.e. it is a Banach space.
Item (I1) is giving me trouble, because I don't know much about tensor products of Banach spaces. In particular, in order for $x^*\otimes y$ to belong to $\mathcal{J}(X,Y)$, it must first belong to $\mathcal{L}(X,Y)$. But, the way Ryan defines tensor products, $x^*\otimes y$ is an element of $\mathcal{L}(X^*\times Y)^{\#}$, where $\#$ denotes the algebraic dual. In other words, $x^*\otimes y$ is by definition (in Ryan's book) a linear functional (not necessarily continuous) from $X^*\times Y$ into the underlying field $\mathbb{K}$. Unfortunately, it is not obvious to me, given that definition of elements of the form $x^*\otimes y$, how to view them as lying in $\mathcal{L}(X,Y)$.
Problem.
There must be a characterization of which I am unaware, which, in the context of (I1) above, allows us to view the $X^*\otimes Y$ as a linear subspace of $\mathcal{L}(X,Y)$. What is this characterization? Does he just mean $(x^*\otimes y)(x)=x^*(x)y$?
Your interpretation of $(x^\ast \otimes y)(x) = x^\ast(x)y$ is the correct one. Ryan explains several different interpretations of tensors in section 1.3 of his book, among which the one intended here.
With this interpretation, the algebraic tensor product $X^\ast \otimes Y$ is simply the space of finite rank operators from $X$ to $Y$. In particular, condition (I1) means in more classical terms that $\mathcal{J}(X,Y)$ must contain all finite rank operators together with the requirement of a natural estimate on the norms of rank one maps.