As you may know both Amazon Prime & Amazon Multi-Channel Fulfilment are carriers, but what you may not know is the difference between the two carriers.
Amazon Prime is a yearly subscription service granting access to "One-Day Delivery on millions of items". All listings on the Amazon Marketplace bearing the Prime logo are available to Prime members with free or heavily discounted, same-day, next-day or premium delivery services. It's no surprise that listings in the Amazon Marketplace displaying the Prime logo tend to get preferential positioning in category listings and help draw the prospective buyer's eye.
Amazon Multi-Channel Fulfilment:
Amazon Multi-Chanel Fulfilment offers the ability and opportunity to take advantage of Amazon's massive storage, packaging and worldwide dispatch services. In the simplest terms, you send Amazon your products and Amazon takes over storage and fulfilment of your orders for you for a fee.
The secondary advantages are that this automatically enables next day and, in some places, same-day delivery options as delivery is undertaken by Amazon's delivery network. As such, your listings will also be Prime enabled and state that fulfilment is performed by Amazon.
What's the catch? Well, there are, of course, fees to pay for each sale and dispatch, and there are both short and long-term warehousing fees dependent on how much stock you send to Amazon to store and how much is sold or remains in warehousing. Amazon recommends keeping around 6-weeks worth of stock available. Interestingly, however, you don't need to be selling any of these products on Amazon's Marketplace to take advantage of these services, so this is an option available to anyone. Simply send your stock to Amazon and complete an online form to place a Fulfilment order or upload a CSV file of shipment data and Amazon will take care of the rest.
Finally, since your order management is passed to Amazon, customer services and refunds/returns are also undertaken by them. If you don't mind others making decisions on returns and replacements, damaged stock and failed deliveries, this could be the choice for you. Amazon are a very customer-first company and will use any inventory made available to them to keep their customers satisfied.