Here's an excellent article that was either written by or copied/pasted by some member here (ElectricBugaloo?). It's a bit long, but worth the read, and thanks to whoever took the time to write this...
Hardcore has its roots obviously in the punk movement of the 1970's, it has been argued that hardcore was a sort of "electric folk" music- but I suppose that applies more directly to punk which was much more strictly political initially- in any event, as has been said before, hardcore happened after punk had gotten faster and louder, and bands discovered they didn't even need to sound "punk" anymore, and started utilizing different song structures and different chord progressions and styles.
Hardcore died in the early 80's because of the Emo movement in part, but also directly from the act of Ian Mackaye disbanding Minor Threat(mistakenly labelled a "straight edge" band for the song of the same name) and forming the band "Embrace". From then on- bands within the hardcore genre that were newly forming were often called "post-hardcore", referring to a new wave of hardcore bands after the initial death of it- just as you hear many new wave style bands being referred to as a "post-punk", and even fugazi is called this sometimes.
So now we have a general understanding of 80's post hardcore- this term carried forward into the 90's with bands like Quicksand, fronted by Walter Schreifels, who had been the lead singer of a number of influential cornerstone bands of the hardcore movement- especially Gorilla Biscuits and Youth Of Today(he also had a hand in CIV). When quicksand released their ep and subsequent first record in the early 90's-(the ep actually came before the album Nevermind, and is often regarded as being more influential to modern rock, though not nearly as recognized by average listeners)- their style was referred to as post hardcore, which was also a way to describe the massive "jump ship" mentality of old hardcore legends who had become turned off by the whole boots, crew cut and hoodies youth crew nazi mentality that was springing up. 80's Emo was the polar opposite of this- a direct slap in the face to the rise of ultra masculine Straight-edge and Youth Crew style hardcore bands.
About Emo
By now, anyone who doesn't know about Rites Of Spring and Embrace and doesn't understand that they signified the death of traditional hardcore has been living under a rock- or only reading Spin and Rolling Stone's bogus and misinformed interpretations of the term "Emo".
Rites Of Spring was started by Guy Picciotto, Embrace by Ian Mackaye- and these two later became the founding members of Fugazi. These bands did not refer to themselves as emo, nor were they responsible for the term. Legend has it that someone yelled that out at a Rites Of Spring show to the general laughter of the audience, and that it was used as a term of derision against the wimpiness of emo compared against hardcore.
For so called "Emo bands", the hardcore movement had turned into a letdown, and they were tired of the ultra macho swaggering and posturing of traditional nyc hardcore bands- where the hardcore "uniform" often consisted of shorts, hoodies, doc martin boots and crew cuts- the emo "uniform" was quite similar to the style of dress Weezer adopted(which is part of what led to the initial confusion over whether or not Weezer had any connection to emo music) in other words: black square framed glasses, dark scraggly hair, sweaters, t-shirts a few sizes too small, tight fit jeans or slacks- ties, black dress shoes, scarves, etc. etc. This look has carried over into today, although now it has become a hipster badge of honor, and you'll see homogenized versions of this outfit all over every profile on makeoutclub.com, not to mention the impact it had on punk music- which is a different story altogether.
In any case- emo music died when the initial wave of emo bands broke up, and started poppier minded bands with cheesy appropriations of Iron Maiden riffs(that's a joke)- these bands were all about kitsch, with a strong sense of unrequited love in their lyrics(something that original emo DID NOT STRICTLY ADHERE TO), which further explains confusion over Weezer's involvement or lack thereof in the movement- but this movement preceded Weezer by a good 5 years.
When the 90's rolled around- a third wave of bands were picking up influence from emo, from hardcore, from post-hardcore and from general indie rock and starting their own bands- the first, greatest and most influential of these would be Sunny Day Real Estate, who then came to be known as the "founders" of 1990's emo, though the actual genre they belong too and invented(and which contains bands like Mineral and The Appleseed Cast) would be Post-emocore indie. In other words, they are an indie rock band influenced by emocore music, but have come along after emocore had already died. At the same time- east coast bands were stealing some of the screaming hardcore ideas of earlier groups like Indian Summer and Mohinder, and referring to themselves as "screamo"- this would be bands like Saetia.
In the present day, there are literally no bands with any actual connection to the original emo movement- you can obviously trace fugazi's roots, and a few other random acts here and there- but the majority of scene "legends" are guys who came into the movement after emo had already died- these were second and third-wavers who are now fronting third and fourth-wave bands, ie: Jets To Brazil, The Promise Ring.
As far as the heritage of Sunny Day Real Estate- it was appropriated by groups like Mineral(who later became The Gloria Record), Further Seems Forever(from which comes Chris Carraba), The Appleseed Cast, Pedro The Lion, and to some extent even Modest Mouse. However, in the case of all of these bands- the influence they take from SDRE largely comes from the first two records, Diary(initially released in '93) and the self-titled "pink album"(released in '95, posthumously). When SDRE reformed to make arguably their strongest record "How It Feels To Be Something On" this is arguably the beginning of a new trend in indie rock- which is appropriation Prog rock styles. This was fully realized on SDRE's final album "The Rising Tide"- and later came to further fruition in bands like The Mars Volta, Coheed And Cambria, and even the latest reincarnation of SDRE: The Fire Theft.
As for screamo- Saetia broke up in '99, by which time the hardcore movement had essentially reignited itself- but it had combined with metal in many cases, as the members of a lot of these bands came of age in the "Age of metal": the 1980's and early 90's. Bands like Converge, Norma Jean, Shai Hulud, SkyCameFalling, Blood Brothers all have appropriated metal influence, to the point where they could hardly be called traditional "hardcore", and in fact, hardcore purists will often lump all these bands together along with poppy, completely non-hardcore acts as "emo", where a band like Hatebreed would gladly be considered purely hardcore. As for the current wave of "Screamo" bands- it's a generic term for a generic style of music. Thursday stole their sound from a variety of now defunct new jersey post-hardcore acts, and Taking Back Sunday would be considered derivative of a derivative.