There are three stages of twilight: civil, nautical, and
Civil twilight is the period between sunset and the time at which
the center of the Sun's disk is 6 degrees below the horizon. I
assume it is called "civil twilight" because at the end of civil twilight
it is difficult or impossible to work outside without artificial lights.
You'll probably notice street lights near your home coming on at the end of
civil twilight rather than right after sunset. Depending on your latitude
civil twilight ends a little less than half an hour after your local sunset.
(The Sun moves 15 degrees per hour along the ecliptic, but if you live at
higher northern or southern latitudes the ecliptic is highly inclined.)
Nautical twilight is the period between sunset and the time at which
the Sun is 12 degrees below the horizon. At 12 degrees, it becomes difficult
to find the horizon, which means you can no longer use a sextant to take your
bearings (important when you're trying to navigate at sea). This is a little
more than 45 minutes after sunset.
Astronomical twilight as Webster says runs from sunset until
the Sun is 18 degrees below the local horizon. After this point, there will be
essentially no scattered sunlight contributing to airglow, meaning you'll
have a truly "dark" sky (or as dark as it will get at
least). At research observatories, observations usually begin in
earnest after astronomical twilight, since scattered sunlight just contributes
to noise in your observations. (Aside: astronomers often image the nearly
uniform twilight sky to make "flat fields," to test the
CCD imaging camera response as a function of position
on the chip.) Astronomical twilight ends about an hour and fifteen
minutes after sunset.
Same thing goes for morning twilight, just replace "sunset" with "sunrise" and run things backwards.