While working with an Arduino connected to my PC via USB on a project which also had an external 12V power source something went wrong and my PC turn off and a part of my project burnt.
I've disconnected everything and turned my PC on again. Now, on the side of the PC where the Arduino was connected, the two USB ports work partially: if I connect one device (an hard disk, or a keyboard or an Arduino...) it works. But if I connect a second device the first is automatically disconnected.
Can you tell me what's happened to my PC USB ports?
Is there a solution to fix the problem?
Not really, other than that you seem to have damaged something inside the PC, probably on the motherboard.
It may be worthwhile opening the PC to see what the USB ports are connected to.
If they are USB 2.0, the motherboard may have other, unused USB 2.0 headers, so you could try reconnecting the ports to a different header
If they are USB 3.0+, the motherboard will likely only have one connector, in which case you could replace the motherboard or fit a USB 3.0 PCIe card with a connector for internal (front) USB ports and use that.
I will take a SWAG as I do not have access to the parts and say you did irreparable damage to your PC.
Following PaulRB's suggestion is the best way to repair it. But in any case get yourself a powered USB extension with several ports. The one I use is powered and has switches. the +5 is isolated with a diode.
I moved your topic to an appropriate forum category @ginoplusio.
In the future, please take some time to pick the forum category that best suits the subject of your topic. There is an "About the _____ category" topic at the top of each category that explains its purpose.