After the sun sets the sky remains light for a while. The significance is that observations during astronomical twilight may be impaired by the continuing glow of the sky. After astronomical twilight, the sky is as dark as it can be, and you can start your long exposures of faint objects.
There are various definitions of "twilight": When it is "not completely dark". For civil purposes, it is "dark" when the sun is 6 degrees below the horizon, for ships it is dark enough to see navigation stars when the sun is between 6 and 12 degrees below the horizon. But for astronomers, who need the sky to be completely dark, twilight doesn't end until the sun is a full 18 degrees below the horizon
I see you've already found TimeandDate's series of articles on the different types of twilight but I'll link it here for convenience.