Answer: this is as easy as sending an email.
Like any good online banking institution, PayPal has really made it as convenient and as simple as one could expect. Futhermore, since PayPal is owned by eBay, it integrates very smoothly with eBay auction transactions.
For most eBay purchases:
Most eBay auctions have a "Pay Now" or "Send Payment" link directly on the eBay page in question. If you follow this link, PayPal will fill in the details of the seller and the auction ID number for you. Often, it will also fill in the S & H information as well. All you will need to do is login with your secret PayPal password and email address, and confirm that your shipping address and primary funding source are correct. You add extra notes for the seller (e.g. "please send via US Post"), and the money transfer takes place immediately. A confirmation email will be sent to you, and your bank/credit card will be debited within two days.
Send money by the recipient's email address:
For some eBay auctions, or for personal money transfers, you go directly to Paypal.com and click "send money". You login, and then copy-paste the recipient's email address into the "send money" form. You'll need to add in the details of the transaction, but the process is very straightforward from there. Again, your personal banking information is always concealed from the buyer.
Next: when you've won an auction at eBay...