Moon Phase

As the world turns, the moon goes through phases. Each phase refers to how much of the moon's face is in shadow. As the moon moves through these phases it is decribed as either "Waxing" or "Waning." Waxing refers to the moon as is passes from New Moon to Full Moon, while Waning refers to the opposite. You can check the current moon phase, as well as its percentage, by using the /clock command.

On an interesting side note, the moon's face will be tinted the color of the current day as the days change (i.e. on Firesday the moon will tint red and on Windsday the moon will tint green).

Full Moon
The moon phase where the entire face of the moon is illuminated (90%-100%) When the moon is in this phase, it is rumored that certain effects are improved/degraded: Certain Notorious Monsters will only spawn during a full moon: Some items have effects that are affected by this moon phase: Certain items and key items can only be obtained during this moon phase:
 * Item drops from monsters increase.
 * Fishing results improve (except in Rabao, where this is the opposite).
 * Crafting skill ups are improved.
 * The number of High Quality items synthesized is decreased.
 * Gardening yields are higher. (rumored to be 150% normal yield)
 * Chocobo Digging is more accurate.
 * Haty
 * White Coney
 * Horror Head - Enmity -50
 * Luna Subligar - Dexterity +2, Agility +2, Charisma +7
 * Selene's Bow - Ranged Attack +5, Ranged Accuracy +25
 * Moongate Pass
 * Ark Pentasphere
 * Moonlight Ore

Gibbous Moon
The moon phase where most of the moon's face is illuminated.

Half Moon
The moon phase where the face of the moon is halfway in shadow, and halfway in light. Also known as "first quarter moon" when the moon is waxing and "last quarter moon" when the moon is waning. Some items have effects that are affected by this moon phase:
 * Selene's Bow - Ranged Attack +15, Ranged Accuracy +15
 * Amood - Occasionally attacks twice (First Quarter Moon only)
 * Amood +1 - Occasionally attacks twice (First Quarter Moon only)
 * Dark Amood - Occasionally attacks twice (First Quarter Moon only)

Crescent Moon
The moon phase where most of the moon's face is obscured.

New Moon
The moon phase where the entire face of the moon is obscured. When the moon is in this phase, it is rumored that certain effects are improved/degraded: Certain Notorious Monsters will only spawn during a new moon: Some items have effects that are affected by this moon phase:
 * Item drops from monsters decrease.
 * Fishing results in Rabao improve (elsewhere, this is the opposite).
 * Crafting skill ups are decreased.
 * The number of High Quality items synthesized is increased.
 * Chocobo Digging is more accurate.
 * Bendigeit Vran
 * Black Coney
 * Horror Head II - Enmity +50
 * Inferno Sabots - INT +1, Dark Magic Skill +3
 * Selene's Bow - Ranged Attack +25, Ranged Accuracy +5

Links
Tayana's Moon Phase Cycles