What is Tyrael?

Tyrael is a walking, talking Deus Ex Machina.

There is no precedent or explanation in lore other than 'Angels are powerful and they can turn one of their own into a mortal by ripping off his wings because that would be cool, OKAY?'

This is Blizzard we're talking about. There are tons of plot holes, and very few of them make sense. The best we can hope for is a retcon at some point in a piece of fiction or an expansion that makes a ham-handed attempt to explain things, introducing seven more complications in the process.

LessPop's answer pretty much summed it up, but I'll also add a few further conundrums:

1. As an Angel, Tyrael was a being composed of "vibrations of light" and only took on the appearance of a human, never actually becoming one. How then does he "rip off his wings"? Bad vibrations I guess...
2. Tyrael can not be Nephalem or human, because he is part of a race that spawned the Nephalem the ancestors of humans. That would be like being his own grandfather.

So Tyrael being mortal, doesn't make him human - and it isn't clear if his appearance is still just an illusion, or a true representation of his actual form.

At the beginning of D3, he rips his wings off and falls from heaven. After this we know that he has human appearance and that he is mortal. There isn't much discussion about his mortality, we know it can happen.

Is he still an angel?

1. He didn´t become human, it is never stated like that. Is just appereance.

   Angels have taken human form in the past, such as Inarius and The Prophet. Tyrael just made the process permanent.1

   Tyrael makes this permanent by detaching his wings, see below.

2. We know that Inarius's wings were tore off by Mephisto (he remains an angel and immortal). Also, he is condemned to be tortured for eternity. Part of the torture consists of removing his eyelids. Angels don't have eyelids, the removal of the wings triggers the transformation from light and sound to flesh and bones.

3. The enchantress Eirena has this dialogue with the player:

   Well, I did not expect you to keep company with a seraph.

   Seraph is the collective name for archangels and angels.

4. If he were Nephalem or human he would also have no "fate" as Ithereal says and would be free to kill Diablo himself.

TL;DR

He is a mortal angel.