Last year Barnes & Noble rolled out their Nook eReader with some unique features, including the ability to loan books to people for two weeks. Today, Amazon joined Barnes & Noble with this capability as they try to catch up with the functionality of the Nook. You can check out the Amazon help page for all the details on loaning Kindle ebooks.
As you can see on that help page, this is still a process under development as it appears you can only loan and accept loans for Kindle book through your Amazon Kindle webpage with no loan initiation or acceptance from within the Kindle devices or various Kindle applications.
The mechanics of the system work similar to the Nook system where you can offer books up for lending and the recipient has seven days to accept, they can borrow it for up to 14 days and you cannot read it during this time, and if the borrower has the book or is in a country without rights to the book you will get the book back in your library after the seven day period ends.
Websites have launched for the Nook to facilitate loaning books and I imagine we will see the same thing happen with the Kindle as well. I am still more of a Nook fan with support for free public library books.