This is very broad question, and the right answer depends a lot on your particular robotic application.
Most users run ROS on 64-bit Intel (x86_64) or ARM (Cortex-A series) with some variety of Linux, most commonly Ubuntu 14.04 or 16.04 (ROS provides pre-compiled binaries for those versions of Ubuntu)
Most of the compatibility issues that people have are not with the processor itself, but rather with graphics cards, wireless (common troubles on most version of Linux), and with their robot-specific hardware such as sensors and motor controllers.
The Ubuntu community has lots of great resources for working through the more common issues with graphics cards and wireless, and Google is a great tool for searching and finding ROS drivers for specific robotics hardware.
Without knowing more about your application I can't provide any more specific guidance.