While colonization of the Moon is not currently possible, it is feasible as a future human endeavor. With sufficient advancement in space flight capabilities, air and waste recycling technology, and solar efficiency, it should be possible to transport building materials for permanent structures on the surface of the moon. Such structures would provide a stable off-Earth location for low-gravity scientific experimentation, possible residential or commercial prospects, or a future "space port" of sorts, in an age where space flight might be common. These things may not be currently possible, but they are certainly feasible in a future that is probably not too far away.
Obviously, if the Moon were to be colonized, we would need to ship in resources as the Moon has none. We could live in small communities, under domes filled with oxygen. Water would need to be shuttled in from Earth. It's not desirable, but it could be a good practice run before colonizing other parts of the galaxy.
Valuable resources, like Helium 3 are on the moon. However Obama killed the Constellation Project which was the United States best chance to get to the moon. Now Russia or China will probably beat America to it and make trillions of dollars selling the resources back to America. I do however believe that someone in the next 50 years will colonize the moon.
There really aren't that many resources on the moon to exploit.
However, a small science-oriented colony could be feasible. Perhaps there are tests that could be performed on the moon that can't be performed on Earth. Naturally this community would require a support base, so then you got people moving there to be doctors, teachers, lawyers,... But it would never get that big unless...
Unless our resource needs were to change entirely which with new technology they could. But new technology and hence new "resources" that are used for it could completely alter the dynamics in a myriad of ways. New technology could turn some oil-rich parts of the Middle East into ghost towns while making Antarctica an attractive place to build a business.
I truly believe that eventually we will live on the moon. With new technology, we can create scientific space bubbles (kind of like Sandy's house on Spongebob). This will provide an oxygen climate for us that we can walk around freely inside. I mean we have no idea what science will have in store for us in the future, it may not happen tomorrow or in a 100 years... But some day, maybe.
In order to live we need atmosphere and the moon has none of these things and neither does Mars. At best the moon with all its minerals and recently discovered water will serve as a source of raw materials from which we can build those colonies most likely the Space Island colonies envisioned by the late Dr. Gerard K. O'Neill both L4 and L5 positions between earth and the moon being excellent locations. Truth is Man must move into space to grow rather than stay on earth and rot as we are now doing.
Human migration is dependent upon resources. The moon does not have any lakes, rivers, and oceans. In fact there is no water on the moon. The most densely populated areas in the world are cities. Most cities are near a body of water usually. There wouldn't be any cities created at all on the moon. At best, the moon might become a research outpost for scientists but not a place for colonization. Mars colonization would make for a more sound debate. The moon will not be a place of colonization, there are no resources on the moon to bring about such a migration.
Colonisation has historically only worked in areas that have natural resources to exploit. The earliest civilised humans settled between rivers, near forests, in rich grazing and farming land, and by fish-rich seas. European colonists settled in the richest places they could find. The moon has no natural resources to speak of, especially not of the kind to support a long-term human settlement. Human settlements have a tendency to grow, because humans reproduce. On the moon, there is no water, no air, no fuel, and no fertile soil. There is no way to support a human population out there. It would be tremendously expensive to even attempt, and we'd gain very little of value that couldn't be gained by robots or just occasional human visits. Even Mars would be more feasible, as it at least has water.