indiemo
A mix of Indie Rock, Post Hardcore, and Emo from the 90's. Died in the early 2000s when bands started mixing it with Pop Punk and Metalcore. It was softer than first-wave emo and had less yelling and the songs were usually 3+ minutes long. It was the most mainstream of emo, with Sunny Day Real Estate the only emo band ever on MTV and some indiemo bands had music videos.
Some good indiemo bands are Sunny Day Real Estate, Texas Is The Reason, The Get Up Kids, and Knapsack.

Jimmy Eat World was indiemo, but they changed to power pop/lame pop rock in the late 90's.

Emo has nothing to do with emo kids, and was around 20 years before term emo kids was invented. Emo kids like metalcore and pop punk, not emo.