There has been a lot of press lately about tablet devices, Apple seems to have set the pace with the iPad and the iPad 2 is rumored to be announced sometime soon.
iPad apps are written primarily in Objective-C using the iOS SDK free from Apple on the iOS Developer Center. It does cost $99 to deploy apps to a device and distribute them however. There are other tools to develop iOS applications such as MonoTouch and PhoneGap
Blackberry is about to release the Playbook which from my understanding runs Adobe Air applications.
HP just announced the Touchpad which runs WebOS, the applications are primarily built using web technologies (HTML, Javascript, etc)
Then there are a slew of devices coming out that run Android, I think the next one that will be available to consumers is the Motorola Xoom. Android apps are developed primarily in Java using the Android SDK which is free and platform independent.
This market is evolving very fast and there are a lot of players, it seems Apple has the solid ground a this point.
Developing applications for these devices can differ quite a bit, one way that you can reach all devices is by developing a tablet centric webapp such as Yahoo!'s tablet site