Wikipedia states that this is a special case of the law of total expectation click me.
Given a partition $A_1,...,A_n$ of the outcome space, we have for a random variable $X$ that $E(X)=\sum_{i=1}^{n} E(X|A_i)P(A_i).$
Now, I know how $E(X|\mathcal{G})$ is defined for a sigma-algebra, but I don't know what the definition of $E(X|A_i)$ is or more precisely, whether we can view it as a special case of $E(X|\mathcal{G})?$
$\Bbb E[X|A_i]$ means $\Bbb E[X1_{A_i}]/\Bbb P[A_i]$--the "elementary" conditional expectation. This notion is linked to $\Bbb E[X|\mathcal G]$ when $\mathcal G=\sigma(A_1,A_2,\ldots,A_n)$. In this case the random variable $\Bbb E[X|\mathcal G]$ is equal to $$ \sum_{i=1}^n 1_{A_i}\Bbb E[X|A_i], $$ and your Law of Total Expectation is related to the Law of Iterated Expectations : $\Bbb E[\Bbb E[X|\mathcal G]]=\Bbb E[X]$.