Conditional Probability vs Multiplication Rule I am not clear about the differences between the conditional probability and the multiplication rule. Both these consist of a probability conditioned to another event(s).  Also, though I thought that the sample space changes only in the case  of conditional probability, here is an example where the sample space changes for the multiplication rule as well. Say, a bag contains 10 identical balls, of which 4 are blue and 6 are red. If 2 balls are drawn at random, the probability of both of them being red is (6/10)*(5/9). Here, the multiplication rule was used and the sample space was altered to 9. Please clear these concepts by using examples for each. 
 A: The "multiplication rule" is a rule involving conditional probabilities:
\begin{align}
& \Pr(A\ \&\ B) = \Pr(A)\Pr(B\mid A) \\
& \Pr(A\ \&\ B) = \Pr(B)\Pr(A\mid B)
\end{align}

the sample space changes only in the case of conditional probability,

That is correct.

here is an example where the sample space changes for the multiplication rule as well. Say, a bag contains 10 identical balls, of which 4 are blue and 6 are red. If 2 balls are drawn at random, the probability of both of them being red is $(6/10)\times(5/9).$ Here, the multiplication rule was used and the sample space was altered to $9.$ 

The sample space was altered because a conditional probability was used:
$$
\Pr(\text{both are red}) = \Pr(\text{1st is red}) \cdot \Pr(\text{2nd is red}\mid \text{1st is red}) = \frac 6 {10} \times \frac 5 9.
$$
A: The multiplication rule is practicized if we are dealing with two independent events and this by calculation of the probability of their intersection:$$P(A\cap B)=P(A)P(B)$$
Dealing with two events that are not necessarily independent we can still find $P(A\cap B)$ by means of a multiplication but this time it is the rule:$$P(A\cap B)=P(A\mid B)P(B)$$
Labeling this also as "multiplication rule" might cause confusion and IMHO must be avoided.

edit (rectification):
I am convinced now (see comment of Michael, and my answer on that) that the muliplication rule is the general rule that: $$P(A\cap B)=P(A\mid B)P(B)$$ which in the special case that $A$ and $B$ are independent simplifies to: $$P(A\cap B)=P(A)P(B)$$
Feel free to unaccept in order to accept another answer to your question.
