The five primary elements of life are carbon (C), oxygen (O), hydrogen (H), nitrogen (N), and calcium (Ca). Together these make up about 98% of the biomass. We also use iron and silicon to make computers and machines. The problem with colonizing the moon is that it has plenty of iron, silicon, oxygen, and calcium, but very little hydrogen, nitrogen, or carbon, making it relatively hostile to life.
Small deposits of hydrogen-containing ice have been observed at the Moon's poles, where NASA plans to set up a permanent base by 2025. Otherwise, trace amounts of H, N, and C come in via the solar wind, asteroids, and comets. For colonizing the Moon on a larger scale - not just using it as a research base - large quantities of these elements would need to be brought to the lunar surface.
Carbon could come from carbonaceous asteroids, which make up 75% of all asteroids, and are extremely rich in the element. Hydrogen could be mined from the poles at first, but might later need to be imported from Earth. Nitrogen would be the most expensive to obtain - it would all need to be imported directly from Earth, unless there are nitrogen deposits under the lunar surface we don't know about, which is unlikely. The good news is that once all the necessary elements are brought over, they could be indefinitely recycled as long as measures are taken to ensure that the elements don't float off into space. Colonizing the Moon would challenging to get started, but once a reliable cycle is formed, our ancestors may forget that it was ever so hard.
Judging by human history and our spirit of exploration, it seems that in the long enough term, colonizing the Moon is quite likely. A feasible colonization project would require launch costs to come down substantially. A number of proposals to decrease the cost of space transit to $100/lb. or less are in the works, although it may be a few decades before they bear fruit. Once they do, colonizing the Moon will not be in the grasp of only governments, but also private entrepreneurs. The Moon could gain an economic foothold by exporting Helium-3, which would be an ideal fusion fuel. Helium-3 is extremely rare on Earth.
One day, we'll look up at the night sky and see city lights on the Moon above us. People on the Moon will look back and see our lights - unless the colonists live on the side of the Moon which never faces us, in which case they'll only see the Sun, the stars, and very faintly, the other planets in the Solar System.