Store a program not connected to the computer

I'm a rookie learning Arduino starting with the basics including reading books on the subject. But I have a question: can I install a program into my Arduino board via the USB cable from my computer and have it stored in the board?
Will the program only run while the board is attached to my laptop via the USB cable? What I would like to do is install the program and use an external power supply to power the board without the board connected to my computer.

Just go ahead an try it, it will work. Once uploaded, the program ("sketch" in Arduino-speak) is permanently stored inside the microcontroller.

olf2012:
Once uploaded, the program ("sketch" in Arduino-speak) is permanently stored inside the microcontroller.

... until you again connect the Arduino to the PC and upload another program.

...R