I have a laptop with Linux OS (Debian or Ubuntu) and a Samsung Galaxy S III mini phone that is connected to internet with Mobile Data. how can i share mobile internet with laptop?
You have to set the phone up as a wifi hotspot then connect your laptop to the SSID of your phone. On my Android phone (4.1.2), I go to Settings, Wireless and Networks then mobile network sharing. Then I can enable the portable wifi hotspot. You can probably get a more detailed answer here: https://android.stackexchange.com/
Typically, there are options under Settings/more settings on Samsung Touchwiz software. There should be the option of wifi hotspot, and probably tethering through USB/bluetooth. Note that typically with Linux, no extra drivers or installation is necessary for the built in USB tethering.
Some carriers block this unless you pay an extra fee, however. In that case, if rooted, you have the option of custom ROMs or extra apps that can give you that functionality, be it through wifi or USB. You can also use adb port forwarding through the client PC's command line. If not rooted, some apps are available that require client software on the PC to connect through USB.