My understanding is that when a positive voltage is applied to the gate in a thyristor, the P region can store excess electrons which can then move to the n region. I don't understand why it continues to function after the gate voltage is removed, and why adding an extra P region makes the current keep flowing in a thyristor whereas in an NPN transistor, no current flows after the gate voltage is removed.
First of all: It is gate current (or base current in the case of a bipolar transistor) and not voltage that matters.
An NPN transistor can be regarded as a current controlled current source. It basically lets a multiple of the current applied on the BE diode pass through CE. It "multiplies" current. Zero current at the base means zero current at through collector-emitter. There is nothing to hold the current at the base.
Now lets look at the TRIAC (for the purposes of this, let's assume that they are the same as thyrisors which they are mostly for resistive loads.)
The PNPN can also be regarded as an NPN and a PNP transistor that are connected in the way shown below (keep in mind that this is exteremely simplified and that in real world application great care has to be taken that leakage currents don't make a DIAC out of the TRIAC.)
simulate this circuit – Schematic created using CircuitLab
When we apply a voltage, no current flows because both transistors block the gate current of each other. If we apply a current on the gate, Q2 begins to conduct which in turn lets current flow at the base of Q1 which in turn lets current flow at the base of +2 even if there is no current anymore applied to the gate.
It's a bit like a self holding relay. A relay is only switched on as long as its coil is energized. Normally, a relay contact falls off pretty quickly after opening the switch.
As soon as the relay is enegized, it gives itself the current to stay energized.
