The RS232 serial port that terminates in the 3.5mm stereo jack is called /dev/ttyS1 in Linux and is hooked up in Serial2 in the Arduino sketch in the latest firmwares (wasn't in the earlier versions).
It's RS232 voltage levels, so you need to check your transformer, but if it's the regular D-Sub 9 you should be able to connect it directly to the jack connector on the Galileo, without any electrical conversion just a slightly odd cable (there is a MAX232x on the Galileo).
On the Galileo Gen 2 you will need to use a MAX232x or similar since the serial port is TTL level and not RS232.
The RS232 serial port is normally hooked up in the console in Linux, but in the latest firmware you can use it for other stuff too.
Hope it helps.