Yes, it is possible to reach the moon in one day (assuming you mean only the time between liftoff and arriving at the moon, not the months of planning, construction, etc. that also goes into such an endeavor). In fact, NASA's New Horizons craft made the trip in around 8.5 hours. The duration of the journey is strongly dependent on how much you want to spend on fuel and whether or not you want to stop at the moon.

The New Horizons was just zipping past on the way to Pluto, so it could be traveling at a very high speed. The trajectories of the Apollo missions were planned so that they conserved energy (fuel), and several involved landing on the moon. Both of these mean that going too fast is undesirable, and contributed to their longer travel times (~3 days each way). Another mission (SMART-1) used an ion engine in combination with gravity assist maneuvers to make an extremely fuel-efficient trip, but arrived more than a year after being launched.

The first moon landing was on 22 July 1969. The last moon landing was on 7 December 1972. Humans have not been back to the moon since.

Since we've gone to the moon in the past, we can certainly imagine doing so again in the future. The reasons we have not been back have mainly been due to politics, e.g. President Nixon, who sent the astronauts to the moon in the first place, also cancelled the program in order to pursue the Vietnam War. To go back, some organization would need to assemble the money, people, and so forth needed to do it. Because technology has advanced since then, going back now would be less expensive than it was in 1969, but still expensive enough that it would require either a government or a very large corporation.