What is Fast boot in HTC One X? I have HTC One X running Android 4.1.1 and Sense 4+. Under Settings &gt; Power I have an option of Fast Boot. The text under it says "Start up and shut down quickly". How will my device be affected if I turn on/off this option?
"Fast Boot" is basically analogous to the hibernate or possibly S3 (STR) sleep functionality on regular PCs (HTC isn't very clear on what precisely it does, but analysis of its claims and action suggests it's one of these).  Rather than fully shutting down and restarting, the OS resumes where it was after a shutdown, allowing the system to get from "off" to "ready to use" much faster.

As mentioned, some apps don't play nice with this, such as the case is in this SO question.

Unless you're regularly needing to shut off your phone for some reason and want it to start back up faster, I'd personally just leave it off to avoid the potential for weird issues.