emo started as a direct offshoot of post-hardcore with more emphasis on emotionally charged vocals and personal lyrics. Everything after that has evolved from the 1st wave of emo, now distinguished with the tag emocore as that's what it was originally called before being shortened to emo, in different ways.
There are 5 main genres of emo. There's emo, emocore, midwest emo, screamo, and emo-pop. The genre started with emocore like I said, which has a very distinct sound, as exemplified by bands like Rites of Spring, Embrace, and Moss Icon. It's a more straightforward sound still closely tied to hardcore, and the main focus is on the aggression and emotional, personal vocals and lyrics.
The next main style is emo, which is a more broad term that kind of encompasses a broader sound than the other genres, and can be used to describe most rock and punk inspired groups with very personal and emotional vocals and lyrics that don't fall under any of its subgenres. This is where you have bands like Brand New, Lync, Jawbreaker, etc.
The next style, and arguably the most popular, is midwest emo, which is kind of a combination of emo and indie rock. The genre is mainly characterized by whiny, strained, shouty vocals, twinkly sounding guitars, and a lot of emphasis on melody. It still definitely retains the experimental nature of post-hardcore though. The genre started with bands like Cap'n Jazz and Sunny Day Real Estate, that were still quite heavy and punk, but has evolved with bands like American Football and Mineral who took the twinkly melodic sound but shed the punk ethos. This then evolved into the next genre, emo-pop
Emo-pop started with bands that had previously made music characterized as midwest emo, but stripped it bare and gave it a very marketable, streamlined sound and structure. This sound is exemplified with bands like The Get Up Kids and Jimmy Eat World, that got really popular and led to the bastardization of the term emo.
