https://www.theregister.co.uk/2018/01/04/intel_amd_arm_cpu_vulnerability/
https://www.theregister.co.uk/2018/01/02/intel_cpu_design_flaw/
Apparently there are two rather fundamental hardware bugs concerning speculative/out-of-order execution which break the memory separation between processes and between a process and the operating system respectively.

Meltdown seems the more exploitable, the more severe but restricted to Intel (and a few ARM? currently anyway) and can be addressed by patches in the OS - those are already in the pipeline for the big three. The drawback of the workaround is a reduction in performance depending on the program.
Spectre is apparently more difficult to exploit but more prevalent (Intel, AMD and ARM!) and more difficult to protect against.