I just studied learning micro controllers and try to understand difference between them and processors. I've read examples of how to use PIC micro controllers to switch on and off LEDs. However, then I found one board PC - https://www.olimex.com/Products/OLinuXino/A13/A13-OLinuXino/open-source-hardware . It's much powerful and I start to think if I can use this one board PC and manage i/o ports for working with LEDs. Do I understand right that 1) we can do it 2) we do it via GPIO connector which fulfil the same tasks as i/o ports in PIC?
Yes, a simple task like turning on a LED should be possible through the GPIO ports as well as the IO ports of a microcontroller. Sometimes there are restrictions on how easy it is to perform certain tasks through a GPIO instead of a IO of the microcontroller. The raspberry Pi for example has more difficulties implementing a PWM than a microcontroller, but it still should be possible.
Simple answer: Yes you can practically almost use them for the same purpose.