Metalcore is a modern fusion of both hardcore punk and heavy metal. Metalcore is known for its emphasis on breakdowns and heavy guitar riffs. Screamed vocals dominate metalcore. Singing is usually present, but the emphasis is on screamed vocals. A lot of bands that are casually referred to as "hardcore" or "metal" are actually metalcore bands. Metalcore has practically taken over the heavy metal music scene.
Screamo is one specific derivative form of hardcore punk. It has nothing to do with heavy metal. It denotes a fusion of hardcore punk and emo, generally featuring emotional lyrics, transitions from soft to heavy passages, and screamed vocals accompanied by more melodic singing. The word screamo is misused all the time. People often call any music with screaming in it "screamo". But screamo is actually only one specific genre. The genre itself is still very popular. There are lots of modern screamo bands that more-or-less still meet the same basic stylistic requirements of screamo.
Post-hardcore is a direct derivative of hardcore punk (hence the title post-hardcore). It simply defines any modern hardcore punk with high levels of experimentation. I know you didn't ask about this genre, but I felt it appropriate to mention. Post-hardcore represents a very wide range of musical styles. Post-hardcore bands may take influence from many other styles of music, and are often closely associated with alternative rock or punk. What identified post-hardcore, though, is the inclusion of hardcore punk elements like screamed vocals. If a band uses screamed vocals and heavy instrumentation, but is not heavy enough to be considered metalcore, then the band will often be labeled post-hardcore. 90.205.53.223  ericdeaththe2nd90.205.53.223