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I recently found out that x86/x64 processors have a maximum temperature defined in their spec sheet. In the spec sheet the "maximum temperature" is referred to as the "Core Tjmax".

For example on an Intel Core i7-620M you can see the spec sheet here if you search for "Core Tjmax" you will see that this model has a maximum temperature of 105°C.

What happens to an x86/x64 computer when its processor reaches "maximum temperature" or "Core Tjmax"?

Short answer:

The CPU asserts the PROCHOT pin and turns itself off.

It is up to the motherboard to act (or not) on that signal.

Longer answer

When the CPU gets hot (but has not yet reached emergency shutdown temperature) it will throttle itself. That means it will lower the multiplier and the voltage. As a result processing get slower but the CPU also generated less heat.

If that fails and temperature rises to then it will reach a point where the CPU asserts the PROCHOT pin and it does a full shutdown of the CPU.

More details can be found on intels site. Look for your specific CPU and select the correct datasheet. ( E.g. This one for some of their mobile CPU's)

AMDs x86/amd64's chips do something similar. (As do all modern CPUs)

The processor will throttle itself (PROCHOT# is signaled) once you reach that temperature. The motherboard may or may not choose to shut the machine down at this point, but there's generally still a chance to gracefully shut down. The operating system may choose to do so when it detects an overheating condition (Linux will shut down with Critical temperature reached (### C), shutting down.). However, some motherboards may simply power down right away, perhaps even before this temperature is reached.

If you go too far past that point, the THERMTRIP# signal is asserted and the processor halts all execution. The motherboard is expected to turn the machine off immediately. From the Skylake-S datasheet, page 86:

Reaching the maximum operating temperature activates the Thermal Control Circuit (TCC). When activated the TCC causes both the processor IA core and graphics core to reduce frequency and voltage adaptively. The Adaptive Thermal Monitor will remain active as long as the package temperature remains at its specified limit. Therefore, the Adaptive Thermal Monitor will continue to reduce the package frequency and voltage until the TCC is de-activated.

Page 89:

PROCHOT# (processor hot) is asserted by the processor when the TCC is active. Only a single PROCHOT# pin exists at a package level. When any DTS temperature reaches the TCC activation temperature, the PROCHOT# signal will be asserted. PROCHOT# assertion policies are independent of Adaptive Thermal Monitor enabling.

Page 90:

Regardless of enabling the automatic or on-demand modes, in the event of a catastrophic cooling failure, the package will automatically shut down when the silicon has reached an elevated temperature that risks physical damage to the product. At this point the THERMTRIP# signal will go active.

Page 109 says that the critical shutdown point for Skylake-S is about 130 °C at the junction.

The other answers go into what happens technically - though they're missing a few minor things that's probably worth posting as part of an answer.

"TJ" probably refers to a thermal probe, likely a thermocouple,(with the J standing for junction) though it might be a thermistor. Essentially there's a small thermometer in your processor, and your processor will throttle down when it overheats.

Its been almost a decade but its worth looking at what happens if the CPU didn't throttle or shut down to protect itself. AMD didn't and the processors would start smoking and cook themselves - going up to 300 degress celsius. While there's a 'maximum' safe temperature your processor should throttle down and/or shut down well before that