"Local Time" represents the time of the reader and the local times given on my web site should be no more than about 30 minutes off for most of the people in the world.
The Lengthy Explanation:
Earth is divided into 24 time zones (for the sticklers out there, I know there are really 28, but the extra 4 are unusual). As you move westward, each time zone you enter is one hour earlier than the time zone you are leaving (until you get to the International Date Line, but that's another story). As an example, if you were driving west through the state of Kansas in the Central Standard Time zone of the United States and you were about to enter the state of Colorado in the Mountain Standard Time zone you would have to set your watch back one hour. So, if your watch said 3:30 p.m., you would need to change it to 2:30 p.m. Because of these time zones the rotation of the Earth allows some sky phenomena to occur at essentially the same time (+/- 30 minutes) no matter where you are on the planet. This basically includes the times of when certain constellations will rise above your horizon.
Here is another example. Time zones span quite a large distance and there is a reason why I noted above that the times are essentially accurate to within +/- 30 minutes. Let's say you are on the eastern border of the Central Standard Time zone. Sunrise occurs at 5:00 a.m. For people further west, however, the sun has not risen yet. Why? Because Earth is round and sunlight does not curve around Earth's limb. The Earth continues to rotate and in the middle of the Central Standard Time zone the sun will rise at 5:30 a.m. More rotation and then the western edge of the Central Standard Time zone sees the sun rise at 6:00 a.m. You can see here that, if this were to continue, sunrise would occur at noon over the Pacific Ocean and at 6:00 p.m. for people in Asia, but this does not really happen because of the time zones. If you were standing on the western edge of the Central Standard Time zone and walked or drove westward into the Mountain Standard Time zone, you would have to set your watch back one hour. So now a person standing on the eastern edge of the Mountain Standard Time zone sees the sunrise at 5:00 a.m. The cycle essentially repeats through every time zone.
So, whenever you see the term "local time" on this web site this basically means your time.