TITLE: When to use Iterated Forcing
QUESTION [6 upvotes]: Iterated forcing is useful for separating cardinal characteristics, and to prove the consistency of Martin's axiom. My question is, what is it about these problems determine that a normal forcing proof can't work and that we require multiple forcings? 
What about a consistency result determines that it need be proved through iterated forcing rather than ordinary forcing?

REPLY [6 votes]: Maybe I can add a natural example to Asaf's great answer. In the case of establishing Borel's conjecture, iterated forcing naturally comes up in the construction. Borel's conjecture states that all strong measure zero sets of reals are countable (it is not important what that means). As it turns out, this statement is independent of ZFC. It is not too hard to construct a counterexample to it using CH. To show that BC is consistent, you need forcing. But how do you force BC? Richard Laver found a poset (now called Laver forcing) that is easy to describe and has the following property:
If $V[g]$ is a generic extension of $V$ via Laver forcing and $A\in V$ is a set of reals which is strong measure zero in $V[g]$, then $A$ is countable (both in $V$ and $V[g]$). (Note that the evaluation of the statement "$A$ is strong measure zero" may change from $V$ to $V[g]$). Now you have made some progress, there are no "old" counterexamples to BC anymore. But there still might be new counterexamples! It is now clear that one should try to iterate this process to make the new counterexamples vanish in the limit. This turns out to work, however it is not enough to iterate this just $\omega$ or $\omega_1$-many times, it is important to do it $\omega_2$-many times (otherwise CH might still hold in the extension). Moreover one has to choose the support correctly, in this case it is countable support for technical reasons.