Where does this have application? I understand how the demand function may be arrived at using the utility maximization problem but I don't understand where the indirect utility function is used and I think the two are somewhat related... Please excuse me if this is a very fundamental question. I have only started Intermediate Microeconomics.
Recall that if $x$ and $y$ are consumption bundles, $u(x)$ is the consumer's utility function, and $u(x)>u(y)$ means the consumer strictly prefers bundle $x$ to bundle $y$.
The indirect utility function $v(p,w)$ is the highest value of the utility function when $u$ is evaluated over all affordable bundles given $(p,w)$. In other words, $v(p,w)=max u(x)$ subject to $p\cdot x \le w$.
So the indirect utility function tells you the consumers preferences over price-wealth pairs: $v(p,w)>v(p',w')$ means the consumer gets to consume a strictly better bundle when prices are $p$ and wealth is $w$ than any other affordable bundle when prices and wealth are $p'$ and $w'$.
This is my opinion. If indirect utility function is important, that is because of Duality theorem, envelope theorem.
Indirect utility function is one of a measure for consumer welfare. This is an optimized value from UMP. UMP is based on ordinal utility function, so indirect utility function can't be used for comparing consumer welfare. So if you want to compare consumer welfare, you should use expenditure function(which is based on CMP).
It is very useful in industrial organization, whereby extensive use of discrete choice econometric models is made. See for instance McFadden 1973.