Question on Jech's introduction to Iterated Forcing (What really is $P\ast Q$?) Jech's introduction to iterated forcing is really confusing me. He starts with two-step forcing, but to be honest, even on the first page of Chapter 16, the first definition where he defines $P\ast Q$  I am confused. I'm really not able to move forward until I understand this definition. Can someone break this definition down a little bit for me?
Here's the definition of $P\ast \dot{Q}$ that Jech provides:
$P\ast\dot{Q} = \{ (p,\dot{q}):p\in P\; \text{and}\; \Vdash_P\dot{q}\in\dot{Q}\}$
$(p_1,\dot{q_1})\leq (p_2,\dot{q_2})$ iff $p_1\leq p_2$  and $p_1\Vdash\dot{q_1}\leq \dot{q_2}$
This vaguely looks like product forcing, but I really think it's just the $\Vdash_{P}$ notation, various dots, and really just not knowing what it is isn't helping. Jech explains the $\Vdash_P$ notation, but that isn't really helping.
 A: Let me try and give an example that might help and clarify. Consider the following scenario. We start with a model of $\sf GCH$, say $L$ for good measure, now we want to violate $\sf GCH$ on some of the $\aleph_n$s. This essentially defines a real number, but you want to be clever, and not use any of the real numbers already in the ground model.
So, as a start you're adding a Cohen real, $c\subseteq\omega$, then you force with $\prod_{n\in c}\operatorname{Add}(\omega_n,\omega_{n+2})$ to violate $\sf GCH$ exactly on those $\aleph_n$s whose index was in the Cohen real.
How would you describe this whole construction from $L$, then? There is no "obvious" partial order. But the Cohen forcing has an obvious partial order, and once the Cohen real was added, the next step also has an obvious partial order.
Now, that second step has a name, since it lives in $L[c]$, and so we can use that name and combine it with the Cohen forcing to create what is the iteration forcing in one go.
So, how would that work? Well, as we "progress" further along the Cohen real, we learn more, for example, will $59$ be in the Cohen real? We don't know at first, but then we do. Once a condition decided that, we know that our next forcing will have a factor of the form $\operatorname{Add}(\omega_{59},\omega_{61})$. This helps to "reveal" the second step one bit at a time.
