Who is the Angel of Death in Islam?

What the Quran says

The Quran mentions him directly, and makes clear that he is a servant carrying out an order, not a power of his own:

Other verses say "Allah takes the souls at the time of their death" (Quran 39:42) and that "Our messengers" take them (Quran 6:61). Scholars explain there is no contradiction: Allah decrees it, the Angel of Death carries it out, and other angels help him.

Is his name Azrael?

The name Azrael (or Izra'il) is widely used in Muslim cultures and books, and many scholars allow using it. But since it isn't confirmed by the Quran or a sound hadith, the safest thing is to call him what Allah called him: the Angel of Death. It isn't a big issue, and you don't need to correct anyone who uses the other name.

How does he take the soul?

The Prophet ﷺ described this in a long hadith narrated by al-Bara' ibn 'Azib (Sunan Abi Dawud 4753). For a believer, angels with bright faces come carrying shrouds and perfume from Paradise, and the Angel of Death says to the soul, "Come out to forgiveness from Allah and His pleasure." The soul flows out gently, "like a drop from the mouth of a waterskin." For someone who rejected Allah, the scene is harsh and frightening.

There is also a well-known story that the Angel of Death came to Prophet Moses, who struck him, and Allah then gave Moses a choice before he died (Sahih al-Bukhari 1339). It shows that even prophets meet this angel, and that death comes at the moment Allah has written. See What do Muslims believe happens after death?

Should I be afraid of him?

Feeling uneasy about death is natural, and the Prophet ﷺ himself felt the pain of dying. But for a believer, the Angel of Death brings good news, not terror. The best preparation is simple: keep your prayers, repent often, and think well of Allah. See How do I deal with fear of death as a Muslim?