Which is why i'm thinking the Elements are involved. They seem to be a combination of the three working in 'harmony', along with looodes of power, which I suppose could be attributed to the age of the alicorns (1000+ years old)
Sadly, all we really have to go off of for Alicorns are a lot of fanon material, which is why i'm hoping we get an alicorn episode that actually explains them.
Same i'm going off only what i've seen, heard, and read in various areas with the only thing being confirmed is the term "Alicorn". It was on one of http://sibsy.deviantart.com/ journal posts that she confirmed the crew were calling Luna and Celestia Alicorns. After that we know that they have existed for possibly 1100 to 1500 years since Luna and Celestia used the elements to trap Discord. There are just TONS of scenarios out there imho and until Mrs. Faust or the current crew confirm things we're really up in the air with things.
Here's what I know so far:
1. Episode 1 - Battle took place between Luna (Nightmare Moon) and Celestia about 1000 years before the start of main series. This is further amended in season 2 episode 1 where we see Celestia AND Luna battling discord with the Elements of Harmony in the stained glass window.
2. Celestia has extensive magical books at her disposal since there is the royal Canterlot library that Twilight seems to constantly visit. Again this is seen in Episode 1 where Twilight is looking for a reference guide.
3. Teleportation? - This one i'm quite curious about as we've seen this in episode 2 where Celestia comes in through the window in a beam of light. We also see this in S3E3 when she appears yet again in a beam/ball of light.
4. Power - Celestia didn't seem to break a sweat reversing everypony that was affected by Twilight's (Want It Need It Spell) as she put it. Twilight is a dang strong magic caster so it's not without question to say Celestia is probably 5x as strong as twilight. Only thing that seems to be stronger that Celestia herself is discord though with him being a trickster (and apparently able to travel back in time) it's hard to truly say.
5. Confirmed Alicorns - Luna and Celestia are the only ones being Princesses however Candice will be a new confirmed one.
After that all material is pretty much fanon since there are still some things left unheard/unknown. Only thing that I am unsure of is if Celestia and perhaps Luna can make a Sonic Rainboom or if theirs would be quite different due to their power.