At present there is no application available that you can use to download books from your public library. It is likely though that when Version 2 of the operating system is released, it will include the ability to use programs available for the Android platform. Since Overdrive have an Android app your will be able to download and use it on your playbook. My wife is waiting for this as well.
I read from another playbook user on a different site that you can use this free converter to change epub files to pdf which can the be read on the playbook. I haven't tried it as I'm researching buying a playbook, but thought this might be helpful. If you do try it let me know if it works or not. Thanks. http://www.epub-to-pdf.com/
There are two possible issues. One is whether the file is in .epub or .pdf format, the other is whether it has DRM (Digital Rights Management) applied to it.
It seems there are Blackberry apps for .epub files.
http://forums.crackberry.com/playbook-apps-games-f243/best-epub-reader-726698/ talks about some. I don't know if the name crackberry means they are for a cracked Blackberry only.
Both these threads talk about NON-DRM .epub files.
There are lots of non-drm .epub files for download: including almost all classics that are out of copyright (try project Gutenberg).
Most bought books and library books have DRM applied.
I haven't seen anything that says you can read DRM books on the Blackberry.
There are hints one might be coming.
I would avoid converting to .pdf for reading if at all possible.
pdf format files don't flow to match your screen size and chosen font size, and are a pain for anything apart from their original application in printing.