Top 8, and the rise of "web 2.0"
Myspace (2003) let you pick a "Top 8" friends list and customise your profile with your own CSS, which meant the whole site was a riot of auto-playing music and questionable colour choices — and it was glorious for it. Facebook followed in 2004, starting as a Harvard-only network before opening to everyone by 2006. YouTube arrived in 2005 and turned "watch this video" into a universal sentence.
Around 2004, people started calling this shift "Web 2.0" — less about new technology and more about a new attitude: the web as something you contributed to, not just read. Blogs let anyone publish; Wikipedia (2001) proved a crowd-edited encyclopedia could actually work; tagging let users organise content their own way instead of waiting for an editor to do it, a habit that picked up the name "folksonomy".
Under the hood, a technique called AJAX let pages update part of themselves without a full reload, which is why sites started feeling less like flipping pages and more like using software. It's a small technical shift that made a lot of the "web app" feeling we now take for granted possible.