A thyristor is simply a light-sensitive transistor.
When you say "thyristor" in the context of flashes you're talking about the entire system.
A system which would quench a flash tube's light mid pulse once a predetermined amount of light hit the thyristor.
Flash tubes vary their power by varying the pulse length - the length of time the tube is "glowing".
So, on a thyristor flash unit you typically had either a set of reference cards, or a mechanical calculator (as the math problem to be solved had three variables - aperture, ISO, distance (shutter speed isn't a variable because even at full power the typical electronic flash tube is illuminated for fractions of the shutter open time)
So you dialed in your distance, your ISO, and your aperture, and the calculator (or reference card, (typical X/Y chart with two variables, cards themselves being the third dimension)) told you how to "dial" in the thyristor.
Since the thyristor itself was a dumb static thing (it only knew one light level) you would put a series of ND filters over it, to "fool" it into triggering at different light levels. This is what the calculator helped you choose.
Basically, the effect was to have the flash turn off when enough light _returned_ to the camera/flash (through the window) to indicate properly exposed film.
A seasoned photographer could use tweak this dumb auto mode by saying "hmm, the subject appears 2 stops darker than the backlight, I'll lie to the thyristor by two stops so it doesn't quench the strobe early."
This is what TTL flash metering replaced. Through The Lens instead of "through the flash's eye". More accurate and allowed for multiple "eyes" to be used in combination with Matrix metering and it's ilk.
PPS: A little more explanation: Distance is an important variable to account for (though it may not be immediately obvious) because you need to account for the light falloff on it's way /back/ from the subject.
PPPS: Zoom head flash units (The classic Vivitar 285, for example) had four variables to solve for, as the zoom head position also needed to be accounted for.