In this question I asked yesterday I put forward two interpretations of a statements such as "System X is consistent". (a) we can think of it as saying no finite sequence of applications of logical rules to the axioms of the system will result in a contradiction and (b) we can view it as a proof in some given formal System Y (where Y could or could not be the same as X) of some appropriate formalization of a sentence like "there doesn't exist a proof of 0 = 1 in System X". Taking the example of PA (Peano Arithmetic) we see that in sense (b), its consistency has been proved in various formal systems, such as ZF as well as much weaker systems. Godel's second incompleteness theorem says that we cannot prove consistency of System X in sense (b) in System X for sufficiently strong systems.
Now the question is why would we trust any proof of consistency of System X in sense (b) in System X. The mere fact we are trying to prove consistency suggests we doubt it in some sense, further if we have System X is inconsistent it is trivial to prove that it is consistent in sense (b). So let System Z be some formal system for which Godel incompleteness does not apply. Say there is some proof the consistency of System Z in System Z. Would that proof be expressing anything meaningful? It seems any proof undertaken in System Z implicitly assumes the consistency of System Z and we could not know, if System Z were to be inconsistent, if your consistency proof was subtly exploiting that inconsistency.
Thus we are forced to ask what the implications of 2nd incompleteness are. I see the import of 2nd incompleteness is this: "we cannot bootstrap consistency proofs". That is we cannot take some exceedingly simple system, System A, that we believe we can take to be manifestly consistent, and from that construct a chain of stronger systems B, C, D, ..., PA, ..., ZF, ... s.t. $\vdash_A$ Cons(B), $\vdash_B$ Cons(C), ...
Edit: At Carl Mummert's request I am adding a direct statement of a question. The most pertinent question is: Does $\vdash_X$ Cons(X) have semantic content? That is does proving the consistency of a System X in System X mean anything, P2 argues why I don't believe it does. P3 then endeavors to understand what the semantic content of Godel's Second Incompleteness Theorem is. So a secondary question is: does that analysis make sense, or does 2nd incompleteness say more?