The blue hour comes from a French expression, l'heure bleue, which refers to the period of twilight each morning and evening where there is neither full daylight nor complete darkness. The time is considered special because of the quality of the light at this time of day.
You can go to this site, bluehoursite.com to learn the best time for blue hour photos in your region.
My dilemma is that I can never remember to go outside during those times and take a photo! Here is one attempt on Sunday: