It's much less intimidating when you know what ps does, and what grep means. ps xa generates a report showing all running processes, one process per line. Try running it by itself, you'll see what I mean.
'| grep mysql' feeds that report through the 'grep' command, which will only print lines that have 'mysql' in them. If mysql was running, you'd see a whole bunch of things named mysql, but since it's not, all you see is the one thing running on the system that has grep anywhere in it's name or options -- the program you just ran, 'grep mysql'. Isn't multitasking fun?