What is the difference between a 2 and a 3 solder terminals socket (coaxial jack dc)
The third terminal is like a switch.  When you have nothing plugged in the third terminal is connected to one of the others.  When you plug something in the third terminal is disconnected.
It's primarily designed for automatically switching between battery and external power.
The schematic symbol is:

With nothing plugged in pin 2 is connected to pin 1.  When you plug something in it pushes against pin 1 separating it from pin 2.