It depends on what you want to do. Raspberry Pi and BeagleBoard are fairly similar, but they are VERY different from Arduino.
RPi and BB are essentially small Linux-based computers, and you program them more or less the way you would program a computer. You can even use Java if you want. Downsides are booting time and non-real-time nature.
Arduinos are very simple 8-bit microcontrollers with no OS, so your code will be run on bare metal. They are much simpler, give you real time guarantees, and boot instantaneously (and can be powered off any time).
Which one you should use depends on what you want to do.
If you want to build robots with simple sensors and motors, an Arduino would be more suitable.
If you want to do any kind of audio or video processing, or use USB devices, you'll have to use RPi or BB.
In many complex robotics applications, people would use both - RPi/BB for the CPU-intensive processing, and a bare metal microcontroller for low level real time control of sensors and actuators.
If you are not experienced in electronics, I would highly recommend starting with an Arduino. It's much simpler.