Sub-$200 portable computing is an exciting development. Certainly these methods will work on many larger devices also.
Apple devices are harder to interface due to the lack of standard USB ports. MakerShed markets the RedPark TTL iOS Cable to interface Apple products with the older dock connector. Apple Bluetooth lacks the SPP protocol required to interface with serial devices like Adafruit Bluefruit.
Communicating between Android and Arduino Mega, Leonardo, Micro, and Duemilanove should be straightforward and are listed in the ArduinoDroid board type list. The developer is active so more devices may be added but it would appear to be a 1.05 IDE fork so support for Arduino Due, Yun, and Galileo are probably not expected until a major revision
For Windows tablets, most boards should work but it could take some experimentation.
Programming Adafruit's Trinket and Gemma are currently beyond Android. For Windows tablets, installation of the Adafruit tweaked 1.05 IDE works well. You must also be able to install the USBtinyISP driver. On Windows 8.1, the unsigned driver will probably be very problematic.
There are programs for many operating systems that allow interaction between an Arduino project and a computer via Wifi, Bluetooth, or USB. That discussion would be lengthy enough for a separate tutorial.
To get you started, there are Android programs in the Google Play store if you search for "Arduino".
On a simpler level, nearly all operating systems can open a terminal or command prompt. If the device can open a serial stream over the communications channel, you may be interacting with your Arduino project.