A 'tablet' is a generic term for a portable computer which (unlike, say, a laptop) doesn't have a physical keyboard. (Users type on a keyboard displayed on the screen).
An iPad is a particular (somewhat expensive!) type of tablet, made by Apple. (An
iPod, as mentioned in your post, is completely different. It simply plays music (and/or video) files).
A Playbook is also a particular (and, again, rather expensive) type of tablet, made by Blackberry.
Apple use their own operating system for iPads but they ensure that there are plenty of compatible apps available:
http://www.apple.com/uk/ipad/from-the-app-store/
Blackberry also have their own operating system for the Playbook which, as you've discovered, severely limits the apps available.
Microsoft also produce tablets ('Microsoft Surface') designed to use Windows as the operating system. However they've been slow to catch on and they certainly wouldn't be my first choice in your situation.
Indeed, I wouldn't go for any of the aforementioned devices. The vast majority of tablets use the Android operating system. (They tend to be generally cheaper than everything above and with vastly more apps available). This article guides you towards some of the best:
http://www.techradar.com/news/mobile-computing/tablets/15-best-android-tablets-in-the-world-905504