The moon exerts different amounts of force, or gravity, on the earth, depending on its phase, as evidenced by ocean tides. This force is also thought to affect seed germination by causing a "tide" in the tiny amount of water in the seed, helping it germinate, and also pulling underground water closer to the surface, creating a moister environment for the seed. It also influences root growth, which increases as the moon's pull on the plant decreases. The other element which is supposed to affect plant growth is the changing amount of moonlight. More light is thought to increase the amount of leaf growth.
For example, seeds should be planted so that they will be ready to germinate around the new moon, so plant fast germinating seeds just before or after a new moon. The gravitational pull of the new moon is greater, which aids in seed germination, as explained above. Over the next quarter, increasing light and decreasing lunar pull will promote balanced leaf and root growth. In the second quarter, increasing lunar pull slows the root growth and increasing light will promote extra leaf growth. These two factors should also stimulate seed germination, if any of your "new moon-planted" seeds still have not sprouted. At the full moon, seedlings should be transplanted. This allows the transplants to become extablished during the third quarter, where low moonlight levels and low lunar gravity encourage decreased leaf growth and increased root growth. This is also the time for pruning, as sap will run less. The fourth quarter, with decreasing light and increasing lunar gravity, is a resting period for plants. This is a good time for harvesting.
Many "moon gardeners" have also devised complex systems for planting different kinds of crops during different phases (such as leaf crops following the new moon, and root crops following the full moon) as well as following the signs of the zodiac. For example, plant a moist crop like cucumber under a "Water" sign: Cancer, Pisces or Scorpio. Harvesting should be done under a "Fire" sign: Aries, Leo or Sagittarius.
So does it work? There don't seem to be any controlled studies at this date that can tell us. It is quite certain that light influences the germination of many seeds, and researchers at the Agricultural Research Service's National Soil Tilth Laboratory in Iowa have found that tilling in the complete darkness of a new moon prevents some weed seeds from germinating (although it can't have been easy to till in the dark). However, any light from streetlights, a flashlight or garden lighting would effectively imitate a full moon. And what effect could moonlight possibly have in the far north, where it never really gets dark at night for most of the summer?
What about the gravitational pull of the moon? Does it affect seed germination and root growth? Well, tides are caused by the gravitational pull of the moon affecting the near side of the Earth much more than the far side. However, a "tide" created in a seed whose two sides are mere millimetres apart, is not likely to have any effect at all. In fact, when you stand next to the seed, you are exerting much more gravity onto it than the moon is. Water tables may move with tides, however, creating slight fluctuations in the location of available groundwater. The moon has other effects on the earth, and such things as barometric pressure may also affect plant growth.
It may be worth experimenting with the phases of the moon, but in a short-season climate, I can't imagine waiting for an extra two weeks to plant out my tomatoes because of the moon. I also start many plants indoors, where they won't be affected by the moon's light, and I'm pretty sure that the moon isn't affecting the water table in my seedling tray. If I did plant them in the wrong phase of the moon, I'm sure they would catch up by the next full moon, instead of suffering irreparable harm, as some proponents of the system imply. So give lunar gardening a try, but don't let your moon phase charts take the place of a good weather forecast.