1st in Winter: Old Moon
2nd in Winter: Snow Moon
Last in Winter: Sap Moon
1st in Spring: Grass Moon
2nd in Spring: Milk Moon
Last in Spring: Rose Moon
1st in Summer: Thunder Moon
2nd in Summer: Green Corn Moon
Last in Summer: Fruit Moon
1st in Autumn: Harvest Moon
2nd in Autumn: Frost Moon
Last in Autumn: Long Night Moon
3rd of four in any season: Blue Moon
Oops: forgot to credit it! http://www.fabandpp.org/cotm/moons.htm
There's also another sequence of names of the moon, related to events that occur close to their appearance:

1 - Moon After Yule
2 - Wolf Moon
3 - Lenten Moon
4 - Egg Moon or Paschal Moon
5 - Milk Moon
6 - Flower Moon
7 - Hay Moon
8 - Grain Moon
9 - Fruit Moon
10 - Harvest Moon
11 - Hunters Moon
12 - Moon Before Yule

The third full moon in four is called a Blue Moon. It's the third rather than the fourth, so that the sequence is mainted throughout the year and the final one is still the Moon Before Yule

There are many names for the full moon depending on when in the year it occurs. It has been common in many traditions and parts of the world to name the full moon to identify it as the moon that watches over a particular place in the seasons, and that is why there are so many different names for one simple recurring astronomical phenomenon. Some will list by the month, calling any second moon that falls in a single month a "Blue Moon." Here are some examples:

January: Wolf Moon, Quiet Moon, Snow Moon, Cold Moon, Chaste Moon, Disting Moon, Moon of Little Winter, Old Moon, Moon After Yule.

February: Ice Moon, Storm Moon, Horning Moon, Hunger Moon, Wild Moon, Red and Cleansing Moon, Quickening Moon, Solmonath (Sun Moon), Big Winter Moon, Snow Moon, Bony Moon.

March: Storm Moon, Seed Moon, Moon of Winds, Plow Moon, Worm Moon, Hrethmonath (Hertha's Moon), Lentzinmanoth (Renewal Moon), Lenting Moon, Sap Moon, Crow Moon, Moon of the Snowblind, Windy Moon, Lenten Moon, Crow Moon, Crust Moon, Chaste Moon.

April: Growing Moon, Hare Moon, Seed Moon, Planting Moon, Planter's Moon, Budding Trees Moon, Eastermonath (Eoster Month), Ostarmanoth, Pink Moon, Green Grass Moon, Grass Moon, Flower Moon, Egg Moon, Moon of the Red Grass Appearing.

May: Hare Moon, Merry Moon, Fright Moon, Flower Moon, Frogs Return Moon, Thrimilcmonath (Thrice-Milk Month), Sproutkale, Winnemonoth (Joy Month), Planting Moon, Moon When the Ponies Shed, Milk Moon, Dyad Moon.

June: Mead Moon, Moon of Horses, Lovers' Moon, Strong Sun Moon, Honey Moon, Aerra Litha (Before Litha), Brachmanoth (Break Month), Strawberry Moon, Rose Moon, Moon of Making Fat, Green Moon, Corn Moon, Flower Moon, Dyad Moon.

July: Hay Moon, Wort Moon, Moon of Claiming, Moon of Blood, Blessing Moon, Maedmonat (Meadow Month), Hewimanoth (Hay Month), Fallow Moon, Buck Moon, Thunder Moon, Ripe Moon, Corn Moon, Mead Moon.

August: Corn Moon, Barley Moon, Dispute Moon, Weodmonath (Vegetation Month), Harvest Moon, Moon When Cherries Turn Black, Green Moon, Fruit Moon, Grain Moon, Sturgeon Moon.

September: Harvest Moon, Wine Moon, Singing Moon, Sturgeon Moon, Haligmonath (Holy Month), Witumanoth (Wood Month), Moon When Deer Paw the Earth, Fruit Moon, Nut Moon, Barley Moon.

October: Blood Moon, Harvest Moon, Shedding Moon, Winterfelleth (Winter Coming), Windermanoth (Vintage Month), Falling Leaf Moon, Ten Colds Moon, Moon of the Changing Season, Hunter's Moon.

November: Snow Moon, Dark Moon, Fog Moon, Beaver Moon, Mourning Moon, Blotmonath (Sacrifice Month), Herbistmonoth (Harvest Month), Mad Moon, Moon of Storms, Moon When Deer Shed Antlers, Frost Moon, Trading Moon, Hunter's Moon.

December: Cold Moon, Oak Moon, Moon of Long Nights, Long Night's Moon, Wintermonat (Winter Month), Heilagmonoth (Holy Month), Big Winter Moon, Moon of Popping Trees, Long Nights Moon, Snow Moon, Moon Before Yule.

Some others count the moons in each season and then assign them names. There will be three full moons allotted to each season, so the first moon's name will go to the first full moon; the second moon's name will go to the second full moon; and the third moon's name will go to the last full moon. This means that if there are four full moons in a season, the third is given the name "blue moon," because if there was no "fill in" moon the placement would be out of order, and for instance the "Moon after Yule" might come after Yule! Here are some examples, these vary:

First winter moon: Moon after Yule
Second winter moon: Wolf Moon
Last winter moon: Lenten Moon

First spring moon: Egg, Easter, or Paschal Moon
Second spring moon: Milk Moon
Last spring moon: Flower Moon

First summer moon: Hay Moon
Second summer moon: Grain Moon
Last summer moon: Fruit Moon

First autumn moon: Harvest Moon
Second autumn moon: Hunter's Moon
Last autumn moon: Moon before Yule

Credit to Richard Minutillo (http://www.fabandpp.org/cotm/moons.htm) and A Witch's Cauldron (http://www.pathcom.com/~newmoon/month.htm) for some of my research and whatnot.

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