What does it mean to condition on a variable B in causal models? Given the causal model $A \rightarrow B \rightarrow C$, then $A$ and $C$ are independent conditioned on $B$. What does this mean?
 A: As it is stated in the Wikipedia article on Causal Models:

Conditioning on a variable is a mechanism for conducting hypothetical experiments. Conditioning on a variable involves analyzing the values of other variables for a given value of the conditioned variable. In the first example, conditioning on B implies that observations for a given value of B should show no correlation between A and C. If such a correlation exists, then the model is incorrect.

Therefore "conditioned on B" means "assuming that B takes one of its specific values". 
In the causal model $A \rightarrow B \rightarrow C$, $A$ is an indirect cause of $C$, so C likely depends on A. However, if $B$ is given (i.e., "if we condition on $B$", i.e. if we have a specific value of $B$), then $A$ is unnecessary to explain $C$, hence we would say that "$C$ is independent of $A$ given $B$".
A: You can also look at this probabilistically:
$$
P(C|B) = \frac{P(C,B,A)}{P(B|A)P(A)} \\ = \frac{P(C|B,A)P(B|A)P(A)}{P(B|A)P(A)}
$$
We cancel out like terms and get:
$$
P(C|B) = P(C|B,A)
$$
Therefore, by conditioning on $B$, $C$ and $A$ are independent because the probability is the same regardless of conditioning on $A$. 
However, we could do better and try to put this into English. Another way to say it would be that $A$ provides no information about $C$ when we condition on $B$ because all of the information $A$ provides is contained within $B$.
