In many Star Trek episodes term/concept in phase, out of phase, phase variance, etc. are used many times.
What does phase mean exactly?
Please give as detailed and technical an answer as you can come up with, including Chemistry, Physics, Biology, etc. If you have sources please cite them.
Please keep in mind the three episodes "TNG: The Next Phase", "TNG: Time's Arrow", "VOY:Deadlock", and the issues/answers that may arise from those episodes when answering the question.
Out of phase, from a physics perspective, means you have two waves that are not in sync:
In audio this causes a weird effect, because the sound coming out of the right speaker reaches you at a slightly different time than the sound coming out of the left speaker. Then when you consider the fact that sound is a wave, the pressure can interfere with each other causing all kinds of other phenomenon.
In Donald.McLean's answer, he quoted this:
Geordi: Well, whatever or whoever is there, we're out of phase with it. But we're only talking by a fraction of a second.
Warf: A fraction of a second would make them invisible?
Geordi: A millisecond, a year -- it wouldn't make any difference. If what we're reading is true, then we're occupying the same space but in a different time.
Applying this idea of being out of phase, it appears then that in Star Trek time is a waveform. And, in this particular case, the Enterprise and her crew were out of phase with these other folks. If you treat the Y-axis of the above graph as a coordinate in 3-d space, and the X-axis as time, if the blue line is the Enterprise and the red line is the other party you can see that aside from the two intersection points, they would always be occupying a different time when they are at the same space.
This actually reminds me of an interesting episode where the Enterprise and her crew were being sucked into this wormhole thing, and Deanna Troy was having problems.
As it turned out
They were actually trapped by 2-dimensional beings that were trying to get "home". Turned out that all they had to do was either dive or raise and they were able to escape. They just happened to be on the same 2-d plane as the beings.
TL;DR - Out of phase is technobabble that allows something to be hidden from normal human senses as a plot device.
To the best of my knowledge, no such phenomenon exists in the real world, and so we can only understand it by examining what actually took place in the episode.
From "Time's Arrow Part 1", starting at approximately 14:23:
Data: Captain, the results of my temporal analysis may be pertinent.
Picard: Go ahead.
Data: Geordi, it indicates a synchronic distortion in the areas emanating triolic waves.
Geordi: Well, that explains a few things. How much, Data?
Data: A positive displacement of .004%.
Geordi: Well, whatever or whoever is there, we're out of phase with it. But we're only talking by a fraction of a second.
Worf: A fraction of a second would make them invisible?
Geordi: A millisecond, a year -- it wouldn't make any difference. If what we're reading is true, then we're occupying the same space but in a different time.
So, there is some group of beings and objects that are "out of phase", in that they are a fraction of a second in the landing party's future. As time passes for the landing party, it is also passing for that which is out of phase so that the landing party can never catch up. It's sort of like two cars that are on the same road, traveling the exact same speed and separated by a small distance. As long as the two cars continue to maintain the same speed, the following car will never catch up.
