This facility allows your website visitors to interact with your site and enter a price that they'd be willing to pay for a particular product (and quantity of product).
For example, if you sell a t-shirt for £10, a website visitor can make an offer for £9 for the t-shirt. This offer is then sent as an email to the website administrator. They can choose to accept or reject this offer. If an offer is accepted, the person who made the offer receives an email stating that the offer has been accepted, and they can easily pay for the item at the offer price by clicking a link within the email. Each offer (if accepted) expires after 4 days.
The website owner has the chance to make a counter-offer, and can reply back with an amended price if they feel the original offer price is too low (no different to haggling in a market!). This is sent as an automated email - the site owner just needs to alter the price in the backoffice, and click "Send".
This facility is purely optional and is automatically disabled as a default setting. The site administrator can choose to select individual items that can be offered on, or could choose to put all his or her items on offer (one click of a button).