Before you suggest it, no, this isn't a reference to the scene at the end of The Hangover Part II where Stu, dirty and tattooed, tells his bride's father there's a demon in him. (It's not a Matt Damon reference, either.)
A daemon, according to Greek mythology, is a benevolent nature spirit. In a practical sense, a daemon is the little voice inside your deepest soul that tells you what to do, or what not to do. Not intuition, for intuition is more of a hunch; not consciousness, for consciousness is apparent as it is; not subconscious, for subconscious deals with different tasks.
If you agree with Ralph Waldo Emerson about each person having a life's task to fulfill, your daemon will try to give you hints as to what that task is. For Socrates (According to Plato's Apology of Socrates), it was a voice that warned him what not to do - but never told him what to do. And for Hellenists there were both good and evil daemons.
So there isn't one singular definition for daemon, but one thing people seem to agree on is its divine nature - it's not an actual deity, but something deity-like, a lesser deity of sorts - a guiding spirit.
Now, you don't have to believe in any of this - I'm not suggesting I do - but ask yourself this: do you hear a voice coming from inside you at times, a voice that perhaps warns you against mistakes, or tells you what you should be doing - not with the present moment, but with your life?
This resembles Daniel Kahneman's System 1 thinking in Thinking, Fast and Slow. If you ask yourself if you're happy, or what you would like to accomplish with your life, or whether a certain action is a good idea, something instantly answers your question.
That's your daemon. The question remains, should you listen to it? Should you heed its advice? Or should you continue suppressing and ignoring it? And if you do, what will happen? Will the quality of your life increase of decrease? Where will you be in a few years' time?
Will you have wasted time, or used it to its potential?