that is not at all how things work you must download the file directly to the device, not just access it from your online library. kindle app is like that every thing is stored in your online account library but you sync it to the device and tap the book to download it to the device so you have access to it at all times.
puting every thing on the sd card is best it saves the onboard memory.
my preferred reader app is aldiko from the play store.
can download to your pc then put it on the sd card.

anything you buy from the market should absolutely get installed on your device, you will have to select to install it then go to it in the app list of the device and tap it do the settings.

I have the same tablet and have no problem with being able to read books offline that I've purchased or downloaded for free. Being able to access content from the tablet without being connected to the Internet is something that you should be able to do.

Are you sure that you actually downloaded the file to your tablet? If so, what is the format of the book? You might need to convert it to another format that is readable on Android. For example, a Nook book in PDB format will only be readable on a Nook. Here is a link that explains a bit about Nook and formats that may help:

http://bookclubs.barnesandnoble.com/t5/NOOK-First-Edition-Technical/B-amp-N-eBook-Format/td-p/597581

I recommend you download the free and very excellent ebook management program called Calibre which makes changing formats very easy. It also excels at finding free ebooks wherever they are available on the Internet.

In addition, I believe that there is a free app called Nook for Android that may also help with this problem.

See what you can figure out and then post back reporting success or asking for more help.

Here is a link that might be useful: Calibre

actually even easier would be to install the nook app and read them in the nook reader same for kindle books read using the kindle app.
I don't think they were actually downloaded to the device I think you were probably just reading them from your nook library.

I have a Nextbook 7 tablet and use the Nook app for Android and the Kindle app for Android and I can download to my memory card and read my books offline.

it might be better if you tell us exactly what steps you are taking so we can determine what is happening.

Sorry I haven't responded sooner. I had to go out of town today.
I read through the responses and will go through them again. Today on the road I thought I'd take screenshots (the Samsung is supposed to make doing that easy) and post pictures of what happens. Then I decided to take it to the Geek Squad at BestBuy first, since I've had the tablet less than a week.
The young lady at BB was able to get the nook app to let me read books when I'm not connected to wifi (I don't have 3 or 4g). But as everything with android so far - she got it to work but didn't know why it worked or what exactly she did to make it work. We sat side by side and she tapped through controls and settings and didn't seem to change anything. But it worked. On the way home I pulled over to the side of the road and checked and it was working. When I got home I checked it again and it is still working. I'm still mystified. She's the first to get it to work and I'm very appreciative.
I'm going to install alkido and play with things for a couple days, then repost to update. I hope things are solved.
Several people have said I should keep files on the sd card I installed. A couple people have said I need a file manager app to do that. Any suggestions?
Any other suggestions are greatly appreciated. I'll let you know how it goes.

I have both the Android Nexus 7 and a HP TouchPad (remember the "fire sale"?). On both I have the Kindle app. Amazon sends the book title to my list or shelf, but I must download the book while connected to wi-fi in order to read. Once the book is actually downloaded, I can read it offline. Same for the Nook app. So it seems that you must not have actually downloaded the book contents while online to cause the problem you had. With those apps, no need for any additional apps to get all the books I want to read. Glad it's all fixed now.

I thought that too, and downloaded the books and magazines
over and over. Each time I'd open the app it was like starting from scratch and everything would have to be downloaded again.
What does the nook symbol on a book in the library mean?

have you used the sync feature to have the tablet and online library to sync up and send the books to the device?

With aldiko use epub file books to try it you can go to www.feedbooks.com and pick a free epub book download it from your tablet it should be available in the aldiko reader.

As to file manager I always use Astro file manager from the play store, however you do have one on the device it comes with all versions of android. The file manager is basically the same as the windows explorer on the pc. Look in your app locker for file manager.

The Nook symbol means that it must be read on the Nook reader. Nook books use the epub format, whereas Kindle books use a different format.

However, once you have the book file downloaded, you can usually convert it in a matter of seconds to another format so it can be used on other devices.

Here is a link that might be useful: Calibre

My thanks to all. It's still working.

hald, a few years ago I was as confused as you are right now. I was searching for a different app for everything. However, I found out that there are Android managers for PC (i.e. Mobogenie), which can help me with almost everything. I've used a few different managers. You just need to find the one that suites your the most.