Every single Alicorn in existence has magic that is tied to a specific source, as well as a signature trait that is unique only to them and their offspring. That source is what gives them their magic, and they essentially become the deity of whatever that source may be.
As long as that source exists, the Alicorn will always exist. This is what makes them immortal. Of course, if an Alicorn chooses, they can expend all their magic and infuse their soul to their source. Doing so will cause them to lose their physical bodies, and they become one with the source that they are tied to. They are considered dead, but they do still technically exist.
Celestia and Luna are not native to Equestria. They come from a distant land called Alvira.
In Alvira, all ponies are Alicorn's. And every Alviran Alicorn is born with their cutie marks, their source and role as deity already chosen for them the moment they are born.
Alviran Alicorn's always physically take after their source, hence why the two sisters look so different, despite being sisters. In Alvira, when an Alicorn has reached maturity, they leave and find a land in need of leadership and they assist as much as they can. When Celestia and Luna reached maturity, they left together and found Equestria. A land in desperate need of guidance. So they took the land and the ponies living their under their protection, and eventually become the rulers of Equestria, leading their new subjects to prosperity.
Celestia is the deity of the Sun. Both her magic and her life are tied to the solar entity. Her unique trait is her Phoenix like tail feathers.
Because of this, her magic has an affinity for fire, and she herself is always exceedingly warm. Both in personality, and in physical temperature.
She is the origin of the Sun aligned ponies in Equestria. Their magic attuned to the same source as her own.
Luna is the deity of the Moon. Both her magic and her life are tied to the lunar entity. Her unique trait is her cosmic markings covering her body.
Luna's magic has an affinity for water and the essence of dreams. She can be quite cold, both in personality and physical temperature. But she has a warm side to her that she only shows to those she trusts.
She is the origin of the Moon aligned ponies in Equestria. Their magic attuned to the same source as her own.
She is also the origin of the Bat Ponies. As before her magic appeared in Equestria, there had never been Bat Ponies before.
Major is half Alviran and half Umbrum. However, because Umbrum are essentially a creature made of pure magic, he shares almost all of his genes with his Alviran ancestors.
Major was born the deity of War. His magic has an affinity for shadows and blood, though he rarely uses the latter ability. His unique trait are the Crystal like growths on his tail.
Major has not stayed in Equestria long enough to affect the magic in a way to bring about a new alignment… however that could change in the future.
Equestrian Alicorn's, while similar to Alviran Alicorn's, are also very fundamentally different. They have traits that are unique to them, however they are not there from birth, they appear sometime later in life after they have already ascended. It is unknown whether Equestrian Alicorn's will ever be the origin to any new alignments or races of ponies.
Cadence and Twilight are the first Equestrian ponies to have ascended into Alicorn's.
Like Alviran Alicorn's, the moment they ascended, they have become the deities of the source of their magic. For Cadence, she is the deity of Love, and Twilight is the deity of Friendship.
And also like Alviran Alicorn's, they are able to become one with their source if they so choose.
Cadance is the deity of Love. Her magic has an affinity for the essence of love itself.
Her unique trait is her massive tail feathers. She does not have a normal pony tail at all, her only tail is her tail feathers.
Twilight is the deity of Friendship. Her magic doesn't have one particular affinity, as her magic is powerful in all forms of magic.
Her unique trait is the smaller secondary horn that has grown under her original one. Both are fully capable of using magic.
Then, we have the newest form of Alicorn's, and the one that not as much is currently known about.
Equestrian born Alicorn's.
Flurry Heart is the first ever Alicorn born in Equestria. She was born with a huge uncontrollable amount of magic that caused a lot of mayhem until she was able to get control of it.
The source of her magic was unknown for a very long time. Her magic was just… there. It did not have a source, and aside from her Harmony alignment, her magic didn't seem to have an affinity for anything.
That was until the day she got her cutie mark.
The moment her cutie mark appeared her magic was easily traced back to it's source. Ice.
It was tentatively determined that Equestrian born Alicorn's would become deities just like the other Alicorn's. However they were not born with their rank as deity already given to them, it was instead something that they would earn at the same time they earned their cutie marks. This theory was not proven correct until the second Equestrian born Alicorn however.
Flurry Heart is the deity of Winter. Her magic has an affinity for snow and ice.
She does not currently have a unique trait, however that is likely to change in the near future.
Royal Grove is the deity of Democracy/Politics. His magic doesn't have a specific affinity, however due to his Sun alignment his magic does seem to slightly favor fire and other warm sources.
His unique trait are the yellow markings on his body. They also glow at night, and when he uses his magic.