Exhaustive technical explanation of signing & verification I apologize for such a noobish question and am ashamed that I'm not getting this yet, but I can't seem to put together all of the pieces.  I've read every question I can find on the subject, the Satoshi paper, and bitcointalk.

How exactly is a transaction signed?

How exactly is a transaction verified?  I simply cannot understand how secure transactions signed with private keys that are presumably unknown to all else can be confirmed to be valid and correct without actually knowing the private key itself.  Please show me in detail how this works.

I know this looks like a dupe, but I've read every related question, and I just can't get to that "ah ha" moment that puts it all together.
This should get you started on public key verification: **Digical signature

In essence, the bitcoin address is the public key.  A hash is created of the transaction and this value is then encrypted using the private key that is known only to the owner of the bitcoin address.  Anybody can then use the public key (the bitcoin address) to decrypt this hash value and determine that the transaction has not been modified by creating a hash of the transaction and then comparing it to the decrypted value.  This serves a two-fold purpose: it protects against transaction modification and it proves that the transaction creator has access to the private key for the bitcoin address.