For instance if we have an outcome with 3 classes A,B, and C. Assuming $X$~$Uniform(0,1)$ and $Y$ is as follows
If $ x > 0.6$
$P(Y = A) = 0.4$
$P(Y = B) = 0.3$
$P(Y = C) = 0.3$
Alternatively, if $ x <= 0.6$
$P(Y = A) = 0$
$P(Y = B) = 0.2$
$P(Y = C) = 0.8$
What is the Bayes Classifier and Bayes Error and how does that change with different distributions for x, say if $X$~$Normal(0,1)$
The Bayes classifier chooses the class with maximum posterior probability, i.e. $P(Y=c|X=x)$ and you have all you need; in other words, you have the posterior probabilities for each possible $x$. For example, if $X=0.2$, $P(Y=C|X=0.2)$ is the maximum, and the class is $C$.
The Bayes error you make is the error when you decide incorrectly, based on the Bayes decision rule. For example, $P(X>0.6)=0.4$ if $X\sim U[0,1]$, and if $X>0.6$, we choose class $A$ as our decision. There is $0.3+0.3=0.6$ probability of the correct class being not $A$. Merging the other case yields to following formula, just waiting for the values to be substituted:
$$P(\epsilon)=P(\epsilon|X>0.6)P(X>0.6)+P(\epsilon|X\leq 0.6)P(X\leq 0.6)$$
