In the paper "Binomial Ideals" by David Eisenbud and Bernd Sturmfels, a binomial ideal is defined as follows,
By a binomial in a polynomial ring $S=k[x_1, \cdots, x_n]$, we mean a polynomial with at most two terms, say $ax^{\alpha}+bx^{\beta}$, where $a,b \in k$ and $\alpha, \beta \in \Bbb{Z}^n_+.$ We define a binomial ideal to be an ideal of $S$ generated by binomials, and a binomial scheme (or binomial variety, or binomial algebra) to be a scheme (or variety or algebra) defined by a binomial ideal.
I'm not sure I understand the definition of a binomial algebra. What exactly do they mean by an algebra defined by an ideal?
If $S$ is a polynomial ring and $B$ is a binomial ideal, then $S/B$ is a binomial algebra. (And $\operatorname{Spec} S/B$ is a binomial scheme, and $V(B)$ is a binomial variety.)