I never found any extended lore on the subject so I decided to post here a piece of my own imagination and role play. Of course if any relevant information exists please do correct me.
Also my point of view is from the Guardians perspective. So feel free to add.

Notes on the Ascension
By Fligmont Baery

I am an Ascended. I fight for the Vigil. I fight in their name. And I know we will be victorious and we will free Telara of it's oppressors. And in no small thanks to people like me.
But some mortals ask me: What is it like to be an Ascended? What does it mean?
To understand us, one must understand that there is nothing free. Even the blessings of the gods come at a price.
Here I will try to explain who we are. Or rather "what" we are. And the price each Ascended has to pay.

The Body
Our bodies may or may not be our own. Though I suspect they are not.
I died moments before Aedraxis was defeated. And since then I remember only darkness. I very much doubt my original body, if it actually was my original body in the first place, would have been preserved so well.
Have you ever seen a mage resurrect something? Nasty business. Even immediately resurrected bodies start to decay very fast.
So I assume the bodies we Ascended possess are someone else's. Moreover they were alive when we were placed inside them. A scary idea, isn't it?
So our bodies are just that: Bodies, vessels to contain our immortal souls. We can die just like the rest of the mortals. We feel pain, hunger and we tire.
Although our bodies much more resistant to permanent damage. It is very hard to overcome an Ascended and even harder to kill one. Probably a gift from the Vigil. For if we got a new body every time we died, we soon would have ran out of fresh bodies and the gods would have an army of spirits to fight for them. Not very convenient.
But when we do die, we are sent back to our bleeding and decimated bodies, so we can finish our job. And after that we limp back to the nearest town, to see healers, apothecaries, who then mend our bodies and spirit so we can fight once again in the name of the Vigil.

The Soul
Some say we are blessed. Immortality is very appealing, it seems, for so many!
But our bodies, while hard to destroy, still mortal. And our souls, while immortal as anyone else's, can only be enslaved.
And here comes the greatest catch, of which no Ascended will speak, even if they knew.
We all died at some point. But our souls were denied eternal rest. The Vigil willed it so. And each time we die again our spirits cry for freedom. We see the light but cannot reach it. It is like to be waiting for your love to come back to you, knowing she never will, but being unable to abandon hope. It feels like having a hole in your chest which devours all and yet it still hungers.
But the Vigil send us back to fight on, to be their sword of justice. And the only hope we have is that one day we shall fulfill our end of the bargain, whatever it is, and be granted our final rest.

The Memories
Not many of us would be able to tell you what I had. And not because what I wrote here may be considered a heresy. No. It is because most of our memories are taken from us. We retain only what is necessary for us to be the perfect warriors. The ultimate weapon of the Vigil. And of course, one cannot be expected to be of any use on the battlefield if he or she remembers his or her last moments alive. All of them.
But I remembered. I remembered it all. Trough my studies of planar magics I figure a way to prevent memory loss. How I wish I did not!
But be it as it may, I cannot undo it. I remember now and will remember, every time I perish again, my final moments on the fields of death.