Sometimes you'll hear these terms thrown around by more experienced photographers, and I'm sure there are some out there who haven't the foggiest idea what they mean.
It's very simple, actually. The "Golden Hour" refers to the time immediately preceding, and then immediately following, the setting of the sun. The quality of the light changes dramatically during this time, and creates deliciously warm, inviting tones. The light has a golden color cast to it... allowing you to create dramatic and inspiring images.
I remember first seeing this in the images created by the National Geographic photographers, who I idolized when I was a teenager. They just seemed to have this beautiful golden quality to them... even though I didn't really understand what it was.
Later, when I became a professional, I learned that the Golden Hour was the ONLY time I really enjoyed doing portraiture.
The "Blue Hour" is a bit different... it's the period of time after the sun sets, but before it becomes completely dark outside... the sky just has a beautiful royal blue tint. It doesn't last very long... certainly not an hour... that's kind of misleading... but it has a gorgeous quality to it:
Don't take my word for it, though. Go out this week, and just watch the sky as the sun sets, and beyond. You'll wonder why you never really noticed it before.