Traditional heavy Metal
There are plenty of sub-genres, but there are lots of bands that don't fit any sub-genre and can't really be called anything except "heavy metal". This can be basically anything from the old school British heavy metal invation to the newer stuff coming out today (not to be confused with nu-metal). This is what most people think of when they hear "Power metal" or such terms. Lots of different styles can be considered traditional heavy metal.

Thrash metal
Thrash really started to come into play in the early 1980's. It was in one way, an answer to the more popular "hair metal" bands of the time. Thrash grew out of older metal such as Black Sabbath combined with chunkier riffs, speed, agrresive vocals, and even some punk. Thrash metal is concentrated more heavily on speed and aggression than it's predacessors.

Doom Metal
Where thrash metal concentrates on being fast and aggressive, doom concentrates on being slower, lower and groovier. Many consider Black Sabbath to be the fathers of doom. Typically the music is slow and heavy. Guitars are tuned down and the vocals are usually clean, although growls do slip in on occasion. Also, while death and black metal are generally more aggressive lyrically, doom metal is very meloncholic, depressive and gives an extremly dark vibe. The music can range quite a bit from the extremely slloooooow all clean vocals, to the faster (but still relatively slow) doom/death bands. Check out http://www.doom-metal.com for some good information and a pretty large list of bands.

Death Metal
Death metal is the opposite end of the spectrum from doom metal. There are a few different schools of death metal. Generally speaking though, the music is fast, aggressive, low. The bass line is usually more pronounced than in other metal sub-genres. The vocal style is trademark to death metal. Usually vocals are growled or grunted. Blast beat drums are very prevalent. The most common misconception is that death metal is all about death and gore and killing and the likes. This is most certainly not the case. While many bands do sing about this type of thing, there are tons of great bands that do not limit themselves to lyrics like that. Lyrics can range to include religion, spirituality, fantasy and beyond. The music can range from the very aggressive and brutal (Aborted, Iniquity, Suffocation) to the more melodic (Ebony Tears, At The Gates) to the very technical (Atheist, Cynic).

Melodic death metal
An obvious sub-genre to death metal, it's becoming popular ... it incorporates more melody. What it lacks in brutallity, it more than makes up for in melodic grooves. The vocals are typically not as harsh, there may or may not be as many blast beats.

Black Metal 
Black metal and death metal are quite closely related. While death is focused on the shear brutallity, black metal is focused more on creating atmosphere. The music is typically higher pitched with less of a bass line. Some bands incorporate the use of keyboards or synthesizers. The production is quite often pretty bad giving it a raw or "trOO necro" quality. Tremelo picking abounds, blast beats are not as prevalent as in death metal but are still there quite often. The vocals are generally high pitched shreiked or screamed. The music can range from the "trOO necro" bands (Darkthrone) to the more melodic (Naglfar) to the very sophisticated orchestral atmospheric (Arcturus). A common misconception about black metal is that all the bands are Satanist and sing about Satan. While many bands are Satanic or anti-christian (note the distinct difference), that is not the case with all black metal bands. Lyrics can range from Satanic to fantasy to witchcraft to just about anything. Many black metal bands are very paganistic in ideology and see the christian church as an intruder in their native lands, leading to the very deep seeded hatred for christianity. Unfortunately, black metal also has a sub-genre known as NSBM. National Socialist Black Metal.  Lyrically many of these black metal bands are very racist against not only Blacks or Jews, but anyone of non-aryan decent.

Progressive metal
Prog metal incorporates the general aggression of heavy metal with the progressive rock sound of the late 70's. Generally speaking, the music has lots of different aspects including key and time changes, unique riffing and such. The level of musicianship is typically extremly high. Dream Theater is by far the most popular band in this genre. The music is generally quite technical.

Nu-metal (this genre is not really Metal)
Generally speaking the music structure is very simplistic. Lyrics cover subjects that are close to todays teens (depression, abuse, drugs etc). There are very few guitar solos and the one's that are there are typically very simple and short. Vocals can range from the more metal influenced to a fusion of metal and rap. Guitars are often downtuned or drop tuned. This genre is generally not very popular among listeners of other metal music, but it is the newest craze and therefore very promoted and heavily played on the radio.