The custom price extension for Magento allows buyers to pay an amount of their choosing for products, services, or donations above a minimum set by the admin. It gives admins control over pricing and payments for items without fixed prices like customized products or consulting fees. Customers can enter a custom price and note at checkout above the minimum, and admins can configure custom price settings and validation messages on a per product basis. Order management also integrates with the extension to view custom priced orders.