Q:

Why should I use applicative functors in functional programming?

I'm new to Haskell, and I'm reading about functors and applicative functors. Ok, I understand functors and how I can use them, but I don't understand why applicative functors are useful and how I can use them in Haskell. Can you explain to me with a simple example why I need applicative functors?

A:

Applicative functors are a construction that provides the midpoint between functors and monads, and are therefore more widespread than monads, while more useful than functors. Normally you can just map a function over a functor. Applicative functors allow you to take a "normal" function (taking non-functorial arguments) use it to operate on several values that are in functor contexts. As a corollary, this gives you effectful programming without monads.
A nice, self-contained explanation fraught with examples can be found here. You can also read a practical parsing example developed by Bryan O'Sullivan, which requires no prior knowledge.

A:

Applicative functors are useful when you need sequencing of actions, but don't need to name any intermediate results. They are thus weaker than monads, but stronger than functors (they do not have an explicit bind operator, but they do allow running arbitrary functions inside the functor).
When are they useful? A common example is parsing, where you need to run a number of actions that read parts of a data structure in order, then glue all the results together. This is like a general form of function composition:
f a b c d

where you can think of a, b and so on as the arbitrary actions to run, and f as the functor to apply to the result.
f <$> a <*> b <*> c <*> d

I like to think of them as overloaded 'whitespace'. Or, that regular Haskell functions are in the identity applicative functor.
See "Applicative Programming with Effects"

A:

Conor McBride and Ross Paterson's Functional Pearl on the style has several good examples.  It's also responsible for popularizing the style in the first place.  They use the term "idiom" for "applicative functor", but other than that it's pretty understandable.

