What is melodic death metal? Probably the best metal genre ever. Surprisingly Wikipedia does a great job of summarizing where melodic death metal came from, so I'm just going to copy that here.
Melodic death metal (also referred to as melodeath or MDM) is a subgenre of heavy metal music that combines death metal with the melodious and harmonic elements of the new wave of British heavy metal. The style originated and developed in Sweden (pioneered by At the Gates, Dark Tranquillity and In Flames) and the United Kingdom (pioneered by Carcass) during the early and mid-1990s. The Swedish death metal scene did much to popularize the style, soon centering in the "Gothenburg metal" scene.
There are three albums that are said to be the ones that started it all. These albums are: At The Gates' Slaughter of the Soul, In Flames' The Jester Race, and Dark Tranquility's The Gallery.
If you want to get your feet wet with melodic death metal these are the albums you should start with. They are what modern bands now draw their inspiration from. They are all great from start to finish.
Modern Melodic Death Metal
Here are some newer melodic death metal bands you might also enjoy.
I'll continue to link more bands in the future but I think this is solid start. I'm also going to be posting this on the crit.tv forums.