I thought a reboot of my phone would solve the problem, but guess what, it didn't.
I checked the cable, used different ports on my pc and other computers, used different cables, nothing worked, it only started to charge the battery.
I own this phone for only a couple of weeks, so perhaps this thing needed a repair under guarantee. But of course this would take weeks before having it back, so I called my best friend as help.
Google offered me this link http://android.stackexchange.com/questions/35313/unable-to-connect-galaxy-s3-to-pc-via-usb
Most of the links referred to Windows drivers but I'm not a Windows user. However, scrolling down led me to a very simple solution:
*#7284# to access PhoneUtil
and select PDA in both sections. Reset phone and that's it.
Credits to the user "eidolon" from Australia.
The only thing is that I don't know how I got in this situation... but at least now I know how to solve it.