Alright, let's break it down.

1. When you create a Bitcoin wallet, two keys are generated: a private key and a public key. The private key is your secret number—kinda like your super secret password—that should be kept safe and never shared with anyone.

2. When you make a transaction or sign something with your wallet, what happens under the hood is that this private key is used to generate a digital signature. This signature serves as mathematical proof that the message or transaction was created by someone who has access to the corresponding private key—it's like having an unforgeable digital fingerprint.

3. Now onto where these keys are stored! They can indeed be stored in various places depending on what kind of Bitcoin wallet you're using:

- Mobile Wallets: If you have an app on your phone for handling Bitcoin transactions, then yes, that app stores your private keys directly on the device.

- Computer Wallets: Similarly for desktop wallets—the software application would keep track of your private keys.

- Hardware Wallets: These little gadgets are built specifically to store cryptographic keys securely offline—they're pretty nifty if I do say so myself!

In all cases though remember—keep those privates safe! Don't share 'em with anyone else; they're called "private" for good reason!