Q:

Is it possible to power an orange pi/raspberry pi directly from a computer's power source?

I am looking for a way to put a raspberry pi like computer in an other regular computer. I don't care if this way is a "hack".
The first thing I need to take care of is the power source (5v, 500ma min).
In this conditions, what would be the best way to power my mini computer ?
Also, it would be better if the pi was powered even when the computer is off (the power source stays on)

A:

This is easily done by using the +5SB line on your PC (I have two RPis in one of my Linux machines powered in this very configuration). +5SB is on as soon as you plug your PC into the wall and this standby supply is usually used to power pre-power on indicators and peripherals, LAN card (for WOL), USB for Wake on USB and sleep mode motherboard circuitry.  
The ATX Power supply spec is here: ATX Specification.
While your power supply may be a different size it will have similar specs (usually printed on the power supply) and typically can support at least 2 A from +5SB. 
I cut into a motherboard power cable extension like this: Cable Matters 24-Pin ATX Motherboard Power Supply M/F Extension Cable – 12 Inches.
These cables are readily available and let you plug in and out your RPi if required without hacking your main power supply cable.  
While you can as suggested in the first answer use a standby powered USB port your RPi won't tell them it's a high current peripheral so your output current may be limited by the USB implementation. 
For example, I know of no PCs that will allow a USB peripheral such as a phone or iPad to charge (high current) while the PC is turned off. 

