I want to say "Java Applets", but that feels like trolling.
Really flash had its place for animations, games, and sound/video stuff. There isn't any single alternative, although a lot of things it was used for can be supported directly nowadays. (Things like using the clipboard, etc.)
What tech can do what Flash does and is as easy and approachable as Flash for non-developers?
I mean sure with WebGL and mad computer graphic skills no problem, but with out?