A TVS is a Transient Voltage Suppressor. There are many kinds, typically it works like a "Zener", clamping overvoltages that appear at the gate, but is more robust than a Zener and withstands higher current pulses.
The 3.3 kOhm is placed to ensure safe turn off of the IGBT and to avoid unwanted turn-on due to the capacitive coupling between gate, collector and emitter (fast dv/dt could actually cause unwanted IGBT turn on!)
By the way, which PWM frequency are you using?