Spectre uses a processor's speculative execution capability and branch prediction, combined with similar approaches to those in Meltdown (eg, cache attacks like Evict+Time), to read secrets from a process's memory. The attack involves training a branch predictor to take one path, and then after many iterations, using a branch mispredict to cause speculative execution of code that shouldn't be run. Up to nearly 200 instructions could be run in some instances. Spectre can affect processors from AMD, ARM, Intel, and others—any processor that uses branch prediction, in theory, which is basically every modern processor—but the attack must be tailored somewhat specifically to the hardware, making it more difficult to implement.
Protecting against Spectre is also more difficult. As the whitepaper puts it, "More broadly, there are trade-offs between security and performance. The vulnerabilities in this paper, as well as many others, arise from a longstanding focus in the technology industry on maximizing performance. As a result, processors, compilers, device drivers, operating systems, and numerous other critical components have evolved compounding layers of complex optimizations that introduce security risks. As the costs of insecurity rise, these design choices need to be revisited, and in many cases alternate implementations optimized for security will be required."
In other words, making our processors faster has come at the cost of security, and the only real fix may end up being to slow things down.