If the buyer made the offer through eBay's "Make Offer" button available on some of your listings, and you clicked Accept, then eBay will send a confirmation to your buyer and you should send the buyer an invoice as usual.
Once paid, you ship.
You cannot sell directly to the buyer without going through eBay. That would be a violation of the rules and may get you suspended. The buyer was attracted to your item because of the eBay listing and they have earned their fees.
You need to advise the buyer that if he wants the item, he must make the offer by clicking on the "Make Offer" button on the listing. Once you accept, then a confirmation is sent by eBay, you send an invoice, he pays, you ship.
Good Luck
One may raise the question: "How would eBay know?"
Think about it. The payment would most likely be done through PayPal - owned by eBay.
Now, if the listing is an auction listing, not a Buy-it-Now with the "Make Offer" option, and the offer made by the potential bidder is below the starting price, all you can do - if you really want to sell it below your original starting price - is to lower the starting price and let the potential bidder bid at that price. If someone else comes along at a higher price before the scheduled end of the listing, you get a few more dollars!
An alternative is to lower the starting price even more, add a Buy-it-Now option (must be at least 10% above auction starting price) at the price the potential buyer wants to pay and let him click on the "Buy-it-Now" button. Personally I would NOT recommend that option as you are potentially giving money away.
Another alternative is to cancel the auction listing (end early) and relist it as a fixed price (Buy-it-Now) listing. Once again, I do NOT recommend that alternative for the same reason.
