I can answer in regards to the Los Angeles City public libraries, and I suspect that many libraries in the United States use the same system.
They have a dedicated page explaining the e-media options available to their members.
From this page it seems that there are quite a few companies/subcontracters that they use to handle the transfer of ebooks from the library to a subscriber's account/ereader - Axis360 , Overdrive and Hoopla , for example.
There are more options listed on their e-media page, as well as links to other free, public resources like Project Gutenberg , Bartleby , and the Alex catalog of public texts that don't require LAPL membership.
ebooks.stackexchange.com/users/51/Eyal
If you own a Kindle e-reader AND if you pay the $79 annual fee for PRIME ("free" shipping) at amazon.com, you can make use of the Kindle Owners' Lending Library.
There are currently 476,987 titles in the Lending Library — each month you can check out one of these books.
(This is how I read Catching Fire , book 2 of the Hunger Games trilogy, last year.)
On the first of the next month, you can choose another book (you must first return the previous book).
It's a definite benefit to owning a Kindle, though the number of well-known books (and authors) is limited.
ebooks.stackexchange.com/users/40/Roger_S
It depends on your locale, as others said.
A small library in Armpit, NW may not have that ability.
New York Public Library (one of the premier ones)
most certainly does: http://ebooks.nypl.org/ http://www.nypl.org/ebooks - explains their book sources (including Overdrive - same as London - as well as TumbleBook, 3M Cloud etc..) http://www.nypl.org/ask-nypl/ebookcentral
ebooks.stackexchange.com/users/139/DVK
I live in the USA.
Several places I've lived offer free access to ebooks from local libraries.
A common way seems to be through using Adobe Digital Editions .
This program does the following:
You pair it with your ebook-reading device on your local computer.
The program checks the book out from the library in your name.
The program puts it on your ebook-reading device, and deletes it once your borrowing time is up.
ebooks.stackexchange.com/users/144/Jeffrey DeLeo
The county library for Portland Oregon offers Hoopla for streaming media as well as 3M Digital Library and Overdrive for ebooks.
But, then, we love our libraries!
ebooks.stackexchange.com/users/65/K7AAY
I know Amazon has a program where, if you pay extra, you can borrow ebooks.  Just about everyone in the USA pays taxes to support their local Library.  Are there any programs that allow library books to be checked out, as ebooks?
Can I get an ebook from the library in the United States?
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<united-states><library><loan>
ebooks.stackexchange.com/questions/547
ebooks.stackexchange.com/users/109/James Jenkins
It depends on your local library. Some libraries have free access to ebooks. The answer to this question will depend on your region, but in theory, there are many libraries that offer this service.
The answer to this question will depend on your region, but in theory, there are many libraries that offer this service.