Doom metal was created by Black Sabbath. Doom signifies a slow, plodding tempo.  As in Death, guitars are downtuned and heavily distorted.  Vocals here are either Death metal growls or clean, mournful voices of both men and women.

Classical instruments such as the piano, woodwinds and the traditionally sad string instruments are often employed.
Doom lyrics can deal with death and satanic themes, but predominantly consist of feelings of lost love, tradgedy, depression and mourning. Despite these dark themes, doom is often uplifting and serves well as catharsis for a depressed state.
Many doom songs will ask the question: WHY?
Why did this happen?  Why is she gone? And so on...

FUNERAL DOOM METAL

The ultimate in slow and extremely heavy metal, Funeral Doom takes the lethargic pace of Doom metal and contorts it into a even more torturously slow sound.  The monotany of Funeral Doom can often bring the listener into a trance-like state.   The name of this genre comes from the slow pace of the music, which is compared to the pace of a funeral procession.

The only Funeral Doom I have heard is SHAPE OF DESPAIR.  It was not something I could listen to more than once a year.
MY DYING BRIDE
Epic Doom/Death metal from Britain.  One of my favorite bands despite the cheesy name.  Crushing heavy riffs and one of the best vocalists around make MDB a great way to see what Doom metal is.
Death metal growls and pleading, mournful clean vox are employed to show the two sides of doom: anger and despair.
Now, if you're feeling down, beware listening to this stuff, as it might just make you worse, but for some people it has a healing effect.
Turn Loose the Swans may be the best MSB album ever and The Light at the End of the World is almost as great.  If you dislike growling, Angel and the Dark River has only clean vocals.

AGALLOCH
Atmospheric doom with folky acoustic guitar work.  Songs are very long and some are repetitive.
Harsh black metal vocals and mournful clean vocals provide a cold atmosphere.  There are interludes of complete tranquility and peace which are then broken by harsh metal assaults.
The Mantle is their second release and features greatly improved production over their debut album.