Not very long and I'm back with more questions. I've learned quite a bit about Bitcoin through my pursuits on here and now I have more technical questions. Right now I'm studying the scriptSig of the input portion of a transaction and the scriptPubKey of the output portion of a transaction. I've found some answers and understand aspects but others I don't.

Pertaining to the scriptSig I see the two aspects, public-key and the signature. I a m comfortable with the public-key aspect. When you try to spend BTC you refer back to the output of the previous tx and the program hashes your public-key thus finding the your address and since the preceding output sent the BTC to your address it is verified you have the key. That seems simple to me, now at least.

When it comes to signature portion of it I'm lost. I know that somehow your private-key you provide verifies your public-key through hashes. This makes sense, I understand how that happens. My question is how do you provide your private-key to the input of your tx to process it and find a line-up with your public-key without putting your private-key in the record? Does the process just use the private-key but do nothing else with it? Also I know you sign a hash of the tx and I get how the hash is made from the tx. But what exactly is that signature? Is it a hash of a combination of a your private and public-key? I found this information:

"The script contains two components, a signature and a public key. The public key belongs to the redeemer of the output transaction and proves the creator is allowed to redeem the outputs value. The other component is an ECDSA signature over a hash of a simplified version of the transaction. It, combined with the public key, proves the transaction was created by the real owner of the address in question. Various flags define how the transaction is simplified and can be used to create different types of payment."

here: https://en.bitcoin.it/wiki/Transactions

and I'm trying to understand the italicized part.