How do people measure the distance between the Earth and The Moon? How do people measure the distance between the Earth and The Moon? 
What methods were used? Where can I find the official data?
There are many ways and I'm not entirely sure who you mean with "how do people measure the distance" (does this exclude space observatories like e.g. Clementine, probes currently in lunar orbit like e.g. Lunar Atmosphere and Dust Environment Explorer a.k.a. LADEE, or any other currently available technology?), but one interesting and extremely precise way is by laser ranging, pointing them towards one or more of the retroreflectors that were left on the surface of the Moon by the Apollo landers, in what is known as the Lunar Laser Ranging Experiment. Since they are laid on the Moon's near side that is always pointed towards the Earth, these retroreflectors are available for measurements to any properly equipped researcher, no matter which country they're from.

By measuring the time it takes for the light to reflect back, we can infer distance by simply multiplying the c (speed of light in vacuum) with the time taken between the light signal being transmitted and received and then dividing all of it by two to get a single leg of the distance. I'll let you read the linked to Wikipedia page on LLRE for more information, but if you find that too boring, here's a YouTube video on Mythbusters: Moon Hoax Retroreflectors that explains it from the practical perspective.

As for results, again quoting that same Wikipedia page on LLRE:


  Some of the findings of this long-term experiment are: 
  
  
  The Moon is
  spiraling away from Earth at a rate of 3.8 cm (about 1.5 inches) per year. This rate
  has been described as anomalously high.
  The Moon probably has a
  liquid core of about 20% of the Moon's radius.
  The universal force
  of gravity is very stable. The experiments have put an upper limit on
  the change in Newton's gravitational constant G of less than 1 part in
  $10^{11}$ since 1969. The likelihood of any "Nordtvedt effect" (a
  composition-dependent differential acceleration of the Moon and Earth
  towards the Sun) has been ruled out to high precision, strongly
  supporting the validity of the Strong Equivalence Principle.
  Einstein's theory of gravity (the general theory of relativity)
  predicts the Moon's orbit to within the accuracy of the laser ranging
  measurements.
  


So these distance measurements are so precise they're a lot more interesting and reliable than merely telling us how far away the Moon is. For one, it also blows any Moon landing hoax theories right out of the water. Links to many more official results can be found in references and external links sections of Wiki on LLRE.

There is also a plethora of other ways to measure the distance to the Moon, with different precision and equipment requirements, but I will let others add some other methods in their answers. If however you had some specific "people measuring the distance" equipped with specific measuring devices in mind, don't forget to mention that in your question. ;)