That depends...
Most libraries that I am familiar with use a system called OverDrive Media. With OverDrive, you can "check out" audio books in MP3 format which will work on the iPad. If, however, you want an actual book in electronic format, they use Adobe Digital Editions (ADE) which is a Flash-based system and it will not work on an iPad, iPod or iPhone.
The technology exists to "unwrap" the ADEPT (Adobe Digital Experience Protection Technology) from the books which would allow you to read these books using just about any reader, including iBooks, for as long as you want. This may not be exactly "kosher" and you do so at your own risk and it does require some degree of computer literacy.