we stated a theorem in class:
if X r.v. is $\sigma(Y)$ measurable then X is a function of Y, where $\sigma(Y)$ signifies the sigma algebra of Y.
This is fine. The Professor sometimes states that X becomes a deterministic function, what does this mean? It seems to me that X is still a random variable even if it can be described by an other random variable.
You are right that $X$ can still be a random variable. Take $Y$ to be uniformly distributed among $\{-2, -1, 1, 2\}$, and let $X = 1_{Y > 0}$. Then $X$ is Bernoulli($1/2$) on $\{0, 1\}$ and $\sigma(X) \subset \sigma(Y)$. What your professor probably means is that once $Y$ is known, $X$ is "determined." In other words, $\mathbb{E}[X | \sigma(Y)] = X.$