Death metal evolved from Thrash during the 80’s. Death metal is much heavier than Thrash.   Guitars are downtuned to a rumbling wave of sound.  The music is often difficult for an uninitiated listener to appreciate because of the extreme distortion and wall of sound.
The hardest part to get used to are the vocals.  Death metal vocals are deep, beast-like growls, and are for the most part unintelligible. The term Death metal is derived from the sound of the vocals, which are likened to that of a zombie!
Constant double-bass drumming is required for most Death metal bands and frequent use of loud, crashing cymbals is also common.  While tempo is usually high, it may slow down for headbanging interludes.

Lyrics of this genre often deal, not suprisingly, with death and satanic themes, but often also deal with ancient mythology or even dark fantasy realms.

GOTHENBURG METAL

This is simply a fancy term for melodic Death metal.  Gothenburg is a city in Sweden where this genre was said to have begun.  Vocals here are the same as normal Death metal and guitars are often just as distorted and downtuned, but the music has more of a discernable melody. 
DARK TRANQUILITY
Awesome melodic death metal.  Keyboards are employed once in a while, but are not overpowering.  Powerful and heavy stuff
Their newest release,
Damage Done, is great as well as any of their older ones.

IN FLAMES
Hugely popular Gothenburg metal.  Their style has changed to almost nu-metal in their newest release to the disgust and anger of many fans.  However, their old albums will never change.
Clayman is an excellent choice,as is the The Jester Race.

OPETH
Progressive Death metal.  These guys are amazing.  They contrast harsh growls with angelic clean vocals and crushing heavy riffs with soothing acoustic guitar segments.  They have very long songs,  many clocking over 7 minutes.
Blackwater Park, Morningrise and Still Life are my favorite Opeth releases.
Damnation is a special 'soft' album showcasing just the calm side of Opeth.

BOLT THROWER
A lesser known band that has been around for quite a while.  Simplistic and classic death metal with all lyrics dealing with war.  These riffs will get you headbanging instantly! 
War Master and Mercenary are both awesome.