An interesting view of Doom Metal is that it's not just a subgenre of Metal and cousin to Death Metal, but also the father of very many subgenres in its own right.
These 'sub-sub-genres'-- that's not a technical term-- form two groups: Those that emphasize and exaggerate particular aspects of Doom Metal to extreme lengths- for instance, the Doom Metal subgenre of Funeral Doom mainly focuses on creating extremely ambient music with an air of absolute despair; and those who simply cross a secondary genre with Doom Metal, for instance Stoner Doom bands blend it with the musical style known as 'Stoner Rock'.
The various Subgenres are listed below: it should be noted that, as Traditional Doom has been expressed in Anark's wonderful writeup above, it shall only be mentioned in passing.
Atmospheric Doom focuses mainly on the fact that Doom Metal can be extremely orchestral and ethereal, and often incorporates classical instruments such as violins. It's one of the least heavy subgenres of Doom, and normally features clean female vocals. Many bands who produce this type of metal are influenced by the existence of what is sometimes called 'Beauty and the Beast' metal-- harsh, death metal grunts combined with clear soprano voices. Examples would be Abysmal Grief and Lethian Dreams.
Deathdoom, or Death/Doom as it's often called, is simply a hybrid of Death Metal and Doom Metal: Using the typical aggressiveness of Death Metal combined with the slowness of Doom, it forms an extra-heavy subgenre filled heavy drumbeats and snarled vocals. It currently the most popular form of Doom Metal, and is often seen as a subgenre of Death Metal aswell. Examples of this genre would be Sinister, Winter, and Swallow the Sun.
Drone Doom is a form of metal perfected to be as heavy as possible. Using complicated, long riffs that often last minutes, the distortion in the music-- if it can be termed as such-- is often as important as the notes themselves. Extreme heaviness combined with deep bass make up a lot of the extra work, with vocals often being unintelligable; the vocal style itself varies from clean to tortured gurlging, though it is generally impossible to identify them. Bands that follow this path are Sunn 0))), Khanate, and Burning Witch.
Sometimes called 'Viking Doom', Epic doom is a cross of Doom Metal with elements of traditional heavy metal and Battle/Viking Metal. It's no wonder they came up with a new word to tack on the beginning with those two to choose from. Obviously, they take the influence mostly in lyrics, tend towards saga-length songs, and use hard and heavy guitar at slower speeds than normal for either parent genre. Vocals are normally cleaner than Deathdoom, but often devoilve into the Thrash Shout. Forsaken, Solstice, and arguably, Black Sabbath are good examples for Epic Doom.
Many class Epic as simply part of Traditional, or- for those who class Traditional Doom as a subgenre of Doom- some even see it as a subgenre of Traditional Doom in itself.
Funeral Doom is made to sound like simple misery. Heavy and slow, Funeral Doom is marked especially by the length of it's songs; whereas a normal Doom Metal song may last eight minutes Funeral Doom tracks most often go on for fifteen to twenty mintes-- some go on for even longer. With the focus on atmosphere above all, lyrics are slow, often incidental and growled or hoarsely whispered, and the music that follows them is distorted and often with little to no drumming. Lyrical content is often full of despair. Some examples of Funeral Doom would be Worship, Until Death Overtakes Me, and Thergothon.
A subgenre cross in turn of Drone and Funeral Doom, Torture Doom currently features exactly two bands: Bunkur and Wormphlegm. Featuring large vistas of subdued, slow, and droning music combined with harrowing shrieks, Torture Doom is well named. Both bands have released very few songs, for good reason- Bunkur's 'Bludgeon' lasts an hour and six minutes, and Wormphlegm's 'In An Excruciating Way Infested With Vermin And Violated By Executioners Who Practise Incendiarism And Desanctifying The Pious' lasts thirty-two minutes, almost as long as it takes to say the name.
Whether two bands can constitute a subgenre is a still deeply debated question.
Industrial Doom is simply another hybrid, being a cross between the thudding sounds of Industrial Metal and the slow drear influence of Doom. It often uses techno influences such as elecrical sounds combined with sparse guitar and vocals of any vocal style- though most often the death metal growl, black metal shriek, or the thrash metal shout- to form great discord and dissonance, vying hard away from any trace of melody. An example of this genre would be Phobos of the wonderful Wraith of the Ropes.
Raw and gut-wrenching, Sludge Doom is the combination of Doom and Sludge Metal, an ultrahard subgenre of metal. Using a brutal, harsh beat and varying between extreme speed and extreme lethargy, and makes an incredibly rough and caustic form of music. It is similar to Stoner Doom in many ways, except for the fact that it is very much heavier.
An excellent example would be dISEMBOWELMENT, aswell as eyehategod, Corrupted or Crowbar.
Stoner Doom, or Stoner, is another subgenre that crosses Doom with a separate genre, this time crossing with Stoner Rock. As both Doom and Stoner are slow, Stoner Doom exaggerates this: Stoner Doom's lethargy can result in songs a quarter of an hour or even more in length, and it has the same resultant speed as Funeral Doom (above). The singing style varies, but most Stoner has heavy bass alongside lazy, downtuned guitar, and often plays extremey heavy long riffs. Most the music feels psychadelic, and Stoner Doom is sometimes called 'Psychadelic Doom'. Sleep, who produced the amazing fifty-two minute long Jerusalem-- actually a toned-down version of the sixty-three minute epic Dopesmoker--, are a stone-set example, as are Electric Wizard or (debatably) Cathederal.
Exactly what it says on the tin. Traditional Doom, sometimes now seen as a subgenre of the subgenre it founded (much like Heavy Metal being a modern subgenre of Metal), is slow and dark in style, and tends to be exceedingly heavy. Black Sabbath, are, of course, recommended, as are Saint Vitus and Candlemass.
Undefinable Doom is not a subgenre in itself; it is, instead, a collection of all the Doom bands who, while definately fitting into Doom somewhere, aren't exactly easy to pin into a specific category. Plenty of bands fall into this 'Other' category, such as Esoteric and Khanate.
Many thanks to anamnesis for the info on Sleep's Dopesmoker-- which he insisted was 'imperative'-- and for reminding me to add the 'empty bracket' of Undefinable Doom.