This answer varies widely, depending on which database/web platform the e-book resides on. Some e-books can be "checked out" and downloaded to a reading app such as Adobe Digital Editions. However, many of our e-book vendors limit offline access to a certain number of pages or chapters that can be saved as pdfs.
Here is a list of some of our e-book suppliers and their policies/access instructions:
ProQuest Ebook Central
R2 Digital Library
LWW Health Library
Thieme E-Book Library
For more information see the Florida State University Libraries e-Books Guide.