TITLE: What is the generic poset used in forcing, really?
QUESTION [22 upvotes]: I'm not a set theorist, but I understand the 'pop' version of set-theoretic forcing: in analogy with algebra, we can take a model of a set theory, and an 'indeterminate' (which is some poset), and add it to the theory and then complete to a model with the desired properties. I understand the category theoretic version better, which is to take sheaves (valued in a given category $Set$) over said poset with a given Grothendieck topology (the double negation topology). The resulting topos is again a model of set theory, but now has the properties you want, absent from the original $Set$.
But what is this poset, really? Is it the poset of subobjects of the set you want to append to your theory/model (say a set of a specified cardinality, or some tree with a property)? Is it related to a proof of the property you are interested in? To clarify, I'm not interested in the mechanical definition of an appropriate generic poset, but what it is morally. Bonus points for saying what it 'is' before and after forcing, if this even makes sense.

REPLY [12 votes]: Let me add some view that hasn't really been mentioned above, namely that of Boolean valued models.  Every poset has a completion, which is a complete Boolean algebra.
To each poset we can define a forcing language that essentially looks like first order
logic, only that we use so called names (which depend on the poset and stand for elements of the forcing extension) as constants.
Now the elements of the complete Boolean algebra can be considered as truth values of statements in the forcing language.
Generic filters of the poset and its completion are in 1-1 correspondence to each other.
A filter in the complete Boolean algebra corresponds to a consistent theory in the forcing language.
So, choosing a generic filter for the poset over a model of set theory corresponds to 
choosing a "generic" theory in the forcing language, the theory consisting of 
statements in the forcing language that are true in the corresponding generic extension.
Actually, with this approach it is not even necessary to really go to a generic extension
at all:  If you want to show that CH is not provable in ZFC, all you need to do is construct
a poset so that in the corresponding forcing language all ZFC axioms have truthvalue 1 in
the corresponding complete Boolean algebra and CH does not.
Since everything that follows from ZFC has truthvalue 1 as well, CH does not follow from ZFC.  Now, of course, the computations and arguments you have to do to push through this program are just as difficult as in the case of the usual approach.