This definition applies to modern post-hardcore, not the old school one.
Post-hardcore contains elements of hardcore, punk, and metal, with slightly heavy, fast-paced guitar riffs. Screaming is the main vocal feature, but there are also melodic breakdowns sung in clean vocals at other times, usually in a fairly high voice.
The lyrics are often regarded as what people believe as "emo" nowadays, but if emo really did stand for "emotional", then practically all music out there should be called emo as well, like Britney Spears. The real emo
is actually very different compared to the emo that MTV brainwashed everyone with, but I won't go into depth about that because there are plenty of entries of emo that talk about the real deal.
A lot of people like to associate modern post-hardcore with emo and screamo, especially since some post-hardcore bands selected those as their genres on their Myspace, but that is because Myspace doesn't have anything near post-hardcore as a selection, so selecting emo and screamo are probably the only other closest genres that people would familiarize with post-hardcore (although the real emo and screamo aren't very similar to modern post-hardcore at all). Please don't call post-hardcore "emo" and "screamo". It's post-hardcore.