In the old days, when movies were shot on expensive film stock, people tried to keep the costs down. It was found that the minimum number of frames the human eye and brain need to have the suggestion of fluid motion is 20 frames per second (fps). To allow for some extra headroom, two dozen shots per second or 24fps(1) quickly became the world standard as a good compromise between motion fluency and film usage.
Meanwhile, the development of electricity took place. Transportation of electric power to households worked best with alternating current. The utility frequency, (power) line frequency (American English) or mains frequency (British English) determines the number of power alternations per second expressed in Hz. There was no world standard at first but finally the Americas and Saudi Arabia settled on 60Hz, most of the rest of the world on 50Hz and Japan using both in different regions.
Many electric devices use the utility frequency as a timing source and so did the first electric televisions. Their circuitry was based on analog electronics without the power to make complex calculations so it made sense that their development was based on 25 (for 50Hz) or 30 (for 60Hz) frames per second. It was not possible to broadcast 24fps film to a 50Hz or 60Hz television at the right speed, because 50 or 60 are not dividable by 24. New standards emerged for television(2): 30fps(3) for 60Hz areas in the world and 25fps for 50Hz areas.
Nowadays, it is technically possible to make cameras with more fps for a more fluid motion, especially in scenes with fast moving action. When tested, the target audience did not approve the result, complaining it was unnaturally fluent and lacking that 'film feeling'. Tradition and the 'feeling' made sure that 24fps is still the standard for movie theaters, even in the digital era.
To comply with all these standards, the D800 supports not only 24fps, 25fps and 30fps but also 50fps and 60fps depending on the resolution. So, now you know why there are so many options.
So... what do I choose, 24fps, 25fps or 30fps?
You need to make a good decision before you start shooting. You can't just edit pieces of say 24fps footage together with 30fps. Conversion afterwards is possible with certain software (like Nattress film fx) at a quality cost, but it's better to prevent the problem altogether by choosing a standard for your work.
Unless you have specific cinematographic purposes, do not use 24fps. If you really need 24fps you will know what you are doing and you will not be coming here for advice. If you insist on trying 24fps be sure to check if your editing software supports this before you start shooting your movie.
That leaves 25fps and 30fps to choose from. If your work will be broadcast on television, ask the broadcasting company what they need. Normally they will ask you to choose the fps matching the utility frequency of the country it is broadcast in, 25fps for 50Hz or 30fps for 60Hz. From the above you will understand why.
For everything else, use the fps matching the utility frequency of the country you are filming in, 25fps for 50Hz or 30fps for 60Hz. This has to do with preventing flicker or banding. Light bulbs and fluorescent tubes blink very fast with the speed of the utility frequency. It's almost invisible to the naked eye, but when you are shooting video with a different fps it can become quite noticeable at times.
To sum it up: if you're in the Americas or Saudi Arabia use 30fps. If you're in Japan use either 25fps or 30fps depending on the region. About anywhere else stick with 25fps.
Utility frequencies around the world
The D800 also supports 50fps and 60fps(4) at a maximum resolution of 720p. The can look unnatural when played at the normal speed, but are good to create slow motion scenes. Use 50fps for 50Hz, 60fps for 60Hz and use at half speed in post processing.
Or go to the index of all Nikon D800 video questions.
"Amateurs worry about equipment, professionals worry about time, masters worry about light." (Mike Larson)