their names are

"The Yama"? Should it be, "Yama is a/the great judge"

Yama judges all souls according to what the person did, when they were alive. 

He would be reincarnated (through the act of reincarnation)

Or he would be tortured in the underworld, for a long time, or for eternity (you can't say, eternity or a long time, well I suppose you can, but it sounds better if you say the shorter time first, "long time, or eternity")

they were both human (not humans)

They were both detectives whose duties were to.....

dont' say, "very intimate friends" it sounds homosexual, you could say, "very close friends, almost like brothers".

In their attempts/endeavors to capture criminals, they would.........

Don't use, "it rained cats and dogs" it's too informal, use something like, "it rained heavily/profusely that day". 

the white angel of death decided to stay there until the black angel of death, arrived.

(you can say, "came" too, but I think, "arrived" sounds a little better).

a hat worn by

Very well done MMF.
I enjoyed listening to your story.

Did you write it down and read it? If not, I think it might sound smoother if you do so the next time.

This length of a story is good because I'll be more inclined to try to correct your mistakes, I wouldn't have time to do it otherwise.

You speak very nicely.

Only thing really to work on, is the, "th" sound, "the, they, their".

The point of the tongue should be JUST inbetween the teeth, for the "th" sound.

Well done.

Craig.

the Angels of Death I prefer the "The Black and White Messagers of Hades"

We Chinese believe that after a person is dead, his/her soul will be sent to the arrested to the underground world by two messagers of Hell where it  which is governed by Yama. Like in the mortal world, there are two messengers act like cops for Yama.  angels of death capturing and leading the way to it. They are called the Black Angel of Death and the White Angel of Death. The Chinese term is 黑白無常 literally translated as "the black and white intransience or impermanence".  One is a white tall figure with a tall white conical hat and the other may be the same looking but dressed in black.  Some depictions show the black one to be fat and squat with a equally flat hat.  However, both have long tongues sticking out of their mouths.

There are no angels in Chinese mythology that corresponds to the Western tradition.

Here is the story.
http://melaminefree.antserve.com/Angels%20of%20Death.mp3
Thank you for listening.

Today, I’d like to talk about the two angels of death we Chinese believe. Westerners believe there is a place called Hale, which is governed by Hades. But, we Chinese believe when a person is dead, his soul will be sent to another world underground, unless he is a good monk or a good Taoist priest, he will go to another very wonderful and comfortable world. Under most circumstances, he will be sent to the underground world. This world is governed by Yama. There are two angels serving Yama (阎罗王). Their name is names are  the Black Angel of Death （黑无常） and the White Angel of Death （白无常）. The Yama is a great judge. He judges all souls every soulaccording to what he did when he was alive. If he did has done many good things when he was alive, he would will be reincarnation – reborned reincarnated/reborn (Actually I did not know the word reincarnate. I only knew reincarnation and reborn. Unfortunately, reborn is an irregular verb.) into a human into a very wealthy family. Otherwise, he will be tortured in the underground world for eternity or for a long time. (I said for eternity and suddenly realized some will be liberated after a long time.  So I corrected myself by appending some words.)

Now let’s let me talk about the angels of death. When the two angels of death were alive, they were both humans. They were detectives, whose duties are to capture/arrest suspects (This word was mispronounced.) When they were boys, they swore brotherhood to each other, and they then becamevery intimateclose friends. In an attempt / action to catch / capture a suspect/criminal, they ... … promised to each other that they would at the a bridge. … The White Angel of Death first came to the bridge. Unfortunately, it rained cats and dogs very heavily that day. The river was flooded. To honour his promise, the White Angel of Death decided to stay there until the Black Angel of Death came arrived. He did wait there until the water flooded and drunk him to death he was drown to death. When the Black Angel of Death came arrived, he found out that the White Angel of Death was dead. So, he was devastated. Because he loved his brother very much, so he hanged himself.

Later they were sent to the underground world, which was governed by Yama. And Yama thought they were both very honest and decent men. So, he decided to make them the angels for of Yama. After they got their new job in the underground world, their duties are to capture all dead men’s souls and lead them the way to the underground world. If that man is a bad man, they will bring him to justice as they did when they were alive. On the hat worn (Mispronounced)by the White Angel of Death, it iswritten “welcome to join us (你也来了).” On the hat worn (Mispronounced)by the White Angel of Death, it iswritten “I am capturing YOU (捉的就是你) .”