Supplimental Information:Alicorns, and philosophical musings · 8:28pm Aug 23rd, 2014
What is an alicorn?
The short answer, one that most will give, is 'a winged unicorn.' or unicorn/pegasus. Those who give the issue some thought would add that alicorns have the magic of all the pony races. However, the question itself tends to cloud the real meaning of what it is to be an alicorn.
First, let's take the Two Sisters. Celestia and Luna. Right away we see that alicorn status involves something further than mere magic. Celestia alone is capable of moving the Sun and the Moon, the latter of which is also Luna's duty. This is also shown by their respective Cutie Marks. As shown in the Hearth's Warming Eve, these tasks formerly required a team of unicorns working together. This would indicate that they have a great deal of magic, but in the Canterlot Wedding, Part 2, we see that Celestia 's power has limits, since Queen Chrysalis is just able to overpower her. Based on the preceding premises, it would seem that Celestia and Luna's powers are geared towards moving their respective cosmic objects and for Luna, traversing the world of Dreams in association. Otherwise, they appear to be on par with normal unicorns, as this level of power seldom shows in any of their other endeavors.
Going back to Canterlot wedding, lets take a look at Chrysalis and Cadance. Again we see the alicorn form, but in Chrysalis' case, she has to siphon energy from others in order to use her powers. This, combined with changelings having horns and wings, along with no visible cutie marks, suggests that they are an aberration of sorts. She is barely able to overpower Celestia in a Beam-o-war, but she had to repeatedly siphon from Shining Armor over a period of time beforehand, in order to match the Sun Princess.
Turning the focus to Cadance, we note that the first demonstration of her special talent, A Love spell of some sort, she is able to instantly quell a heated argument between two ponies. In the second demonstration, she reignites Shining Armor's love, and toether, they are able to drive back the entire changeling swarm, Chrysalis included. So it would seem that Alicorns are more powerful than unicorns. The 'but' comes during the Season Three premiere. Cadance has kept a barrier up for several days, and is visibly showing signs of fatigue. So outside of her special Talent, it would seem that she has a limited amount of power.
The newest Alicorn, and the one we get to see ascending, is Twilight Sparkle. Her own ascension during the Finale of season three is where we see, for the first time, what sets an Alicorn apart. A 'Fulfillment of Destiny' As evidenced by the scenes and songs during the episode, Destiny is linked to ones' Cutie Mark. So how did Twilight accomplish this? let's back up a bit. During the three seasons prior, we see that Twilight has a great potential for magic, learning a wide range of spells, and increasing the strength of her magic. We also know that her power is connected to her friendships with the other five Element Bearers. Both factors come into play. Helping her friends overcome a powerful but incomplete spell, and then fixing that spell. 'New Magic' as Celestia states. To flip this around, Twilight could have memorized all the current spells in the Equestrian Library system, and never would have ascended. Rewriting a spell that affected her friends was the key.
To Summarize: Alicorn Ascension seems to be a result of one becoming the very incarnation their special talent.
What is an Alicorn? An Avatar, an ippomorphic incarnation of the ponies' special talent.
This makes sense, though it doesn't actually resolve the differences between so-called "born" alicorns and those who achieve it. (Unless, of course, those "born" alicorns really weren't, and performed their act of achievement at some point lost to history, which is entirely plausible.)