One widespread misconception about the Moon is that it does not have an atmosphere. In fact, it does have an atmosphere, it's just that it has no atmosphere to speak of; it has a mass of less than 50 metric tons, and is spread over a surface area the size of Africa. To give you an idea of just how insignificant that is, a paper published in 1974 estimated that each Apollo mission actually temporarily doubled the size of the Moon's atmosphere1. The reason for the Moon's smoke wisp of an atmosphere is that the sun constantly bombards the solar system with solar winds and ultraviolet light, which ionize gas atoms on contact; this ionization in turn subjects these atoms to the mercy of interplanetary magnetic fields, which send them sailing off into the void (this effect is the basis of the ion propulsion drive, which is far more effective and provides more total velocity than rocket engines.)
In paper cited above, the author also estimated that the creation of an earth-density atmosphere on the Moon could be accomplished by vaporizing oxygen-rich moon dirt with nuclear blasts. Not only is this idea bat-out-of-hell crazy, it would require a nuclear arsenal 10,000 times the size of the U.S.'s current one, the creation of which would be both technologically and politically prohibitive.
Another important aspect of the Moon is that its influence keeps the Earth from wobbling around on its axis, which is set at a very specific 23.5 degrees. It is this orientation that keeps weather patterns stable, i.e. winter during one part of the year and summer during another. If the Moon were magically removed, the Earth could wobble around wildly, causing ice to form where jungles once bloomed and life in general to change beyond all recognition.
1 "Creation of an Artificial Lunar Atmosphere," Richard R. Vondrak, Nature, Vol. 248, April 19, 1974, pp. 657-659-- I found this on a website called MarsInstitute.info, a wonderful little site that has summarised and cataloged hundreds of papers about potential missions to Mars and the Moon. Unfortunately, sporadic interest in space exploration has prevented any of these ideas from coming to fruition