Forget Blackberry. The major players right now are Apple, Google (Android) and Amazon. Microsoft is a recent and promising new entry to the field.
Apple is probably the easiest to use, and integrates really well with the Apple iTunes store for purchasing and downloading music, movies, etc. It also has a large selection of apps. But it has some annoying limitations for power users. I sold mine because I just couldn’t get used to the lame on-screen keyboard design, and Apple wouldn’t allow keyboards developed by third parties. Another drawback is that there are only 2 models to choose from, small and large (7.9" and 10"), plus the choice of different memory capacities (you can’t add memory later).
Android is a mobile operating system developed by Google and licensed to many different tablet manufacturers (ASUS, Samsung, SONY, Toshiba, Lenovo, etc). Google also sells their own Nexus line of tablets (choice of 7" and 10" models), and these are probably the best choices for most people, unless you want particular features offered by other brands. (E.g. ASUS Transformer series transform to a laptop form factor by connecting to a keyboard dock.) Since Android is made by Google, it integrates really well with various Google services - Gmail, Google Maps, Picasa, Google Play Music, Google Play Movie, Google+, etc. Android tablets also tend to be more affordable than Apple.
Amazon’s Kindle Fire series is sort of a limited version of the Android tablet - limited in the sense that you can only get apps from the Amazon App Store, which only has a subset of what’s available for a generic Android tablet. It also has features not found on standard Android tablets, like the ability to download or stream video from Amazon Instant Video (including the “free” Amazon Prime videos.) Kindle Fire tablets are more affordable than other comparable tablets - Amazon can keep the tablet price low because they profit from selling music and TV/movie downloads.
The Microsoft Windows-8 tablets just became available a few weeks ago. The selection of apps is very limited, but the basics are covered. The big advantage of Windows-8 tablets is that they can run Microsoft Office. So if you think you’re going to do “real work” on the tablet, Win-8 is a good choice. (Note that there are two versions of Windows-8, standard and RT. They both run Office but only the standard Windows runs existing Windows software.)
Anyway, if you already have an iPod or iPhone and use iTunes, the iPad is probably the best choice. If you’re a regular Amazon customer (esp. if you’re a Prime member), and if Amazon provides most of the content you want to consume on the tablet (e-books, movies, TV, magazines, etc), the Kindle Fire series is probably the best choice. If none of these factors apply, I think the Google Nexus series are the most versatile and still reasonably priced.