Each month of the year has a different name for the Full Moon. There could be multiple names used, depending on the heritage or system being drawn upon. There are practical, natural world reasons for the names of each of the Full Moons, and they make logical sense when you think about the month and season and what is going on in nature at that time.
The names of the Full Moons follow a traditional pagan way of living, whether that be from North American Native people, indigenous Europeans, or others. They also follow the seasons in the northern hemisphere. I'm not certain if the southern hemisphere has different names for these Full Moons. Please let me know in the comments, if so.
January - Wolf Moon, Cold Moon
February - Snow Moon, Ice Moon, Quickening Moon
March - Storm Moon, Worm Moon
April - Pink Moon, Seed Moon, Wind Moon
May - Milk Moon, Flower Moon
June - Strawberry Moon, Sun Moon, Honey Moon, Mead Moon
July - Hay Moon, Thunder Moon, Buck Moon, Meadow Moon, Blessing Moon
August - Corn Moon, Sturgeon Moon
September - Harvest Moon
October - Blood Moon, Hunter's Moon
November - Beaver Moon, Mourning Moon
December - Oak Moon, Cold Moon, Long Night's Moon
January and February are generally cold and snowy, hence the Cold Moon, Snow Moon, or Ice Moon. In January, hungry wolves could be seen and heard around the villages and civilizations, hence the name Wolf Moon. February marks the end of the harshest part of winter as we start focusing ahead to spring. It is a quickening toward the renewal of spring, hence the name Quickening Moon.
March is often a stormy month, and it has the Storm Moon. This is also the month when earthworms start to be visible again, hence the name Worm Moon.
April is time for seeding gardens and crops, and it is the month when wild seeds start to sprout and grow, so we have the Seed Moon. It is also often a windy month (Wind Moon) and is the month when a pink flower called moss pink or moss phlox starts to bloom again (Pink Moon).
May is a time when the grasses and plants are starting to come back plentifully. Because they have more fresh food to nourish their bodies, livestock start to produce more milk around this time. This is why we have the Milk Moon in May. It's also the month when wildflowers are growing plentifully, so another name is the Flower Moon.
June is when the strawberries are ripe and also when the daylight hours are longest, hence the Strawberry Moon or Sun Moon. It is also called the Honey Moon due to the honey harvest and the Mead Moon from the honey mead that is produced and drunk around that time.
July is haying season, so it is called the Hay Moon. There are often thunderstorms in the thick of the hot summer, hence the name Thunder Moon. New antlers appear on the heads of bucks around this time, hence the name Buck Moon. The meadows are overflowing with beauty and life at this time, hence the Meadow Moon, and the blessings of nature are many, hence the Blessing Moon.
The corn is starting to be ripe in August, so we have the Corn Moon. The sturgeon were most easily caught in the lakes during this time, hence the Sturgeon Moon.
The last three Full Moons of the year (Harvest Moon, Hunter's Moon, and Beaver Moon) occur with the most moonlit hours of any time of the year.
The Harvest Moon (September) produces enough moonlight by which to continue harvesting the year's crops well into the night.
The Hunter's Moon or Blood Moon (October) provides extra moonlight for hunting, to prepare the winter's stores.
The Beaver Moon (November) marks a period of time when beavers are most active in preparing their dens for the winter and can be seen working by moonlight. In this month, we are coming to the end of the fall and beginning of winter, the end of the growing season and beginning of the dark months, and this Moon is sometimes called the Mourning Moon.
December is the month with the most dark hours of the year, hence the name Long Night's Moon. It is one of the coldest months, so it can be called the Cold Moon. Oak trees are symbols of strength, indicating the strength needed to make it through the long, cold winter, hence the name Oak Moon.