When you talk about cartoons, one of the oldest forms of cartoons are newspaper comic strips. Though they no longer be the main source of cartoon entertainment for people today, they are still around, which probably means that it will stay around for a bit more longer, or possibly even as long as any form of print media exist.
Here is a quick tutorial of how these old comic strips are made:
Comics Strip Evolved
One style of comic, I believe, probably evolved from the traditional newspaper comic strip, and that is the 4 panel web comic, showing a complete story through just four panels through simplification of complex subjects, such as emotions or concepts with no form, given often cute forms as characters.
Example: One of my favorite web comic artists.