The vulnerability dubbed “Spectre” affected microprocessors that performed branch prediction as a method of improving system performance when evaluating complex instruction paths run by the CPU. These processors would speculate on the most likely choice when presented with a series of choices. These choices could act on private data and bring this data into cache. A careful observer of access times could use the timing of these actions to infer the contents of the speculatively accessed memory by observing timing results (commonly referred to as a timing attack).
The specific instruction of interest (SWAPGS) is only available on the x86-64 architecture, as such only x86-64 platform vendors (Intel and AMD) are known to be affected.
There is no known complete mitigation other than updating the kernel and rebooting the system.