Meltdown is an exploit that affects Intel CPUs at least since 2011, which leverages elements of out-of-order execution to cause a change in the cache state of a CPU, and then use that to dump contents of memory that should normally be inaccessible. It may affect many other CPUs as well, basically anything that uses OOOE, which includes all Intel CPUs back to the original Pentium Pro (excluding Itanium and Atom before 2013), and AMD CPUs from a similar time period. While the current full implementation of Meltdown does not work on AMD and ARM CPUs, there are indications that further modification of the code could allow a similar attack to work on AMD and ARM processors. The KPTI (and similar) patches that have been deployed for Windows, OS X, and Linux largely mitigate the problem, though there are still some less critical remaining concerns.
It's important to note that many of these exploits aren't actually new. From the whitepaper, "The fact that hardware optimizations can change the state of microarchitectural elements, and thereby imperil secure software implementations, is known since more than 20 years. Both industry and the scientific community so far accepted this as a necessary evil for efficient computing." What has changed is that Meltdown is a working attack vector on many Intel CPUs. The good news is that all the major operating systems should already be patched to mitigate problems.